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The Law of One Price in Equity Volatility Markets
Peter Van Tassel∗
December 2019
Abstract
This paper documents law of one price violations in equity volatility markets. Whiletightly linked by no-arbitrage restrictions, the prices of VIX futures exhibit significantdeviations relative to their option-implied upper bounds. Static arbitrage opportunitiesoccur when the prices of VIX futures violate their upper bounds. The deviations widenduring periods of market stress and predict the returns of VIX futures. A simple tradingstrategy that exploits the no-arbitrage deviations earns significant returns with minimalexposure to traditional risk factors. There is evidence that systematic risk and demandpressure contribute to the variation in the deviations over time.
Keywords: Limits-to-Arbitrage, Volatility, VIX Futures, Variance Swaps, Return Pre-dictabilityJEL Classification: G12, G13, C58
∗Federal Reserve Bank of New York (email: [email protected]): I thank Richard Crump, BjornEraker, David Lando, David Lucca, Lasse Pedersen, Ivan Shaliastovich and seminar participants at theWisconsin-Madison Junior Finance Conference, the Federal Reserve Bank of New York, and the CopenhagenBusiness School for comments. I thank Charles Smith for excellent research assistance. The views expressedin this paper are those of the author and do not necessarily reflect the position of the Federal Reserve Bankof New York or the Federal Reserve System. Any errors or omissions are the responsibility of the author.
1 Introduction
The law of one price is a fundamental concept in economics and finance. While studiesdiffer in their conclusions for how risk is related to return, the vast majority of analysisassumes the absence of arbitrage which implies the law of one price. The law states thatassets with identical payoffs must have the same price. In competitive markets with limitedfrictions, traders will exploit any deviations so that financial markets behave as if no arbitrageopportunities exist.
Given its powerful role for theory and practice, a large literature studies the limits ofarbitrage and law of one price. Challenging the traditional paradigm, well-documentedanomalies illustrate cases in which assets with closely related payoffs trade at significantlydifferent prices for prolonged periods of time. Understanding these anomalies improves ourknowledge of financial markets by identifying the frictions and inefficiencies that preventeven sophisticated traders and institutions from being able to take advantage of seeminglyprofitable situations (Lamont and Thaler (2003), Gromb and Vayanos (2010)).
This paper documents new evidence of systematic law of one price violations in equityvolatility markets. These violations matter because equity volatility markets are amongthe largest and most actively traded derivatives markets in the world. Since the financialcrisis, rapid growth in the S&P 500 index options and VIX futures markets has led to thedevelopment of separate venues where investors can hedge and speculate on stock marketvolatility. These markets provide a testing ground for the law of one price because they offerredundant securities upon which arbitrage pricing places tight restrictions (Merton (1973),Ross (1976a)). In practice, however, trading desks at banks and hedge funds tend to focuson specific products, using different models for hedging and valuing different derivatives(Longstaff et al. 2001). This risk management approach makes it difficult to determine whenrelative valuations and risk exposures are accurate, leaving open the possibility of observingarbitrage violations.
This paper studies relative pricing in equity volatility markets by computing an upperbound for the prices of VIX futures from S&P 500 index options. The upper bound followsfrom a straightforward application of Jensen’s inequality. The payoff to a VIX futurescontract is the difference between the futures price and the value of the VIX index at maturity.Because the VIX index is defined as the square-root of a one-month variance swap, the upperbound for a VIX futures contract is a one-month variance swap forward rate expressed involatility units, not variance units (Carr and Wu 2006). The square-root adjustment in thedefinition of the VIX is convenient because it expresses the VIX in implied volatility unitsthat are familiar to traders who use the Black-Scholes-Merton model, but the definition also
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introduces a wedge between the prices of VIX futures and variance swap forwards. Thepaper applies this observation to define a no-arbitrage deviation measure. The deviationmeasure is equal to the price of a VIX futures contract minus its upper bound which is thecorresponding one-month variance swap forward rate.
The paper finds that VIX futures exhibit significant deviations relative to their upperbounds. Static arbitrage opportunities in which the prices of VIX futures violate their upperbounds occur for 19% of contract-date observations from 2004 to 2018 for the front sixVIX futures contracts. There is at least one upper bound violation on 56% of the days inthe sample. When the upper bound violations occur, traders can sell VIX futures and payfixed in a specific quantity of variance swap forwards derived in the paper to lock in anarbitrage profit. Moreover, these violations are not merely a reflection of measurement errorin estimating variance swap forward rates. Large violations of more than .50% occur lessfrequently but are still present for 5% of contract-date observations. To put this magnitudein context, a mispricing of .50% or half of one VIX-point is more than 10 times that typicalbid-ask spread of .05% in the VIX futures market.
The paper finds that the upper bound is also relatively tight. The average bias andstandard deviation of VIX futures prices around the upper bound are similar in magnitudeand both around .80%. Large deviations below the upper bound indicate that VIX futuresare cheap relative to index option prices. Combining the upper bound with a convexityadjustment from a term-structure model, the paper estimates a lower bound and finds thatVIX futures prices are below the lower bound on 21% of contract-date observations. Takentogether, the evidence suggests that there are large and significant deviations between theprices of VIX futures and their corresponding no-arbitrage bounds implied by the indexoption market.
The paper then explores whether the deviation measure predicts the returns of VIXfutures hedged with variance swap forwards. The results indicate that the deviation measuresignificantly predicts returns across contracts, sample periods, and horizons while remainingrobust to competing predictors like the variance risk premium. The predictability also holdsusing a variety of different datasets to estimate variance swap forward rates, showing thatthe results are not driven by using a particular variance swap dataset. A simple tradingstrategy that sells (buys) VIX futures and hedges with variance swap forwards when thedeviation measure is large and positive (negative) earns an annualized Sharpe ratio of 3.0from 2004 to 2018 with minimal stock market exposure. For comparison, the stock marketearned a Sharpe ratio of .5 over the same period of time as measured by CRSP value-weightedreturns. The large Sharpe ratio from the relative value trading strategy and significant returnpredictability results are consistent with the law of one price interpretation and notion of
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trading against an arbitrage spread.What drives the no-arbitrage violations and return predictability results, the VIX futures
or index options market? Since the analysis is relative, it is challenging to provide a sharpanswer to this question. One hypothesis is that the larger and more established index optionsmarket would serve as a fair-value measure for VIX futures. Alternatively, given the highliquidity in the VIX futures market and the fact that the contracts are traded directly, ratherthan being estimated from option portfolios across maturities, one might hypothesize thatVIX futures would be fairly priced with index option prices driving the deviations.
The paper tests these hypotheses by running return predictability regressions in whichVIX futures and variance swap forward returns are hedged with the stock market returns. Forexample, to the extent that the deviations are driven more by mispricing in the VIX futuresmarket, one would expect larger predictability results for the deviation measure when VIXfutures are hedged with stock market returns as opposed to variance swap forward returns.The results indicate that the deviation measure remains most significant at predicting VIXfutures returns, particularly during the post-crisis period from 2010 to 2018. In contrast,the deviation measure is less significant at predicting the returns of variance swap forwards.This distinction provides some evidence that VIX futures are driving the deviations andpredictability. That said, the strongest results occur when trading VIX futures againstvariance swap forwards, a relative value finding that is silent on the market driving theresults.
Having documented the existence of arbitrage opportunities and the return predictabilityof the deviation measure, the paper then explores what drives the law of one price violationsover time. On one hand, the results are surprising because the index options and VIX futuresmarkets are large and liquid exchange-based markets with competitive traders and limitedshort-sale constraints. On the other hand, the literature on anomalies shows that frictionslike limited arbitrage capital, demand shocks, and financial constraints can lead to returnpredictability and law of one price deviations (Shleifer (1986), Shleifer et al. (1990), Liuand Longstaff (2003), Adrian and Shin (2013)). In addition, asynchronous observations ofVIX futures and index option prices may impact the measurement of the law of one pricedeviations and transaction costs may prevent traders from implementing the arbitrage inpractice. Which, if any, of these hypotheses can explain the observed no-arbitrage deviations?
The paper examines this question by exploring whether a range of variables can explainthe variation in the deviation measure. VAR analysis and panel regressions indicate thatrisk and demand variables are most significant. When stock market returns are negative orwhen realized or implied volatility increases, the deviation measure tends to decline, withVIX futures cheapening relative to variance swap forwards. There is also evidence that the
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deviation measure is increasing in various proxies for VIX futures demand pressure. Oneexplanation for these results is that hedgers take profit on long positions in VIX futureswhen risk increases, leading to demand shocks that are correlated with systematic risk. Atrivariate VAR with the deviation measure, VIX index, and dealer net positions in VIXfutures from the CFTC’s Commitment of Traders (CoT) Report provides some evidence infavor of this hypothesis. To the extent that dealer positions in the CoT report are a veilfor retail demand as argued by Dong (2016), the results are consistent with retail hedgerstaking profit on hedges after risk increases, but also chasing momentum in the VIX futuresmarket as evidenced by increases in long positions after the deviation measure increases.
The remainder of the paper proceeds as follows. Section 2 provides a literature review.Section 3 describes equity volatility markets. Section 4 defines the no-arbitrage deviationmeasure. Section 5 presents the return predictability and trading strategy results. Section6 relates the deviation measure to risk and demand factors. Section 7 concludes. TheAppendix includes additional results and robustness checks.
2 Literature Review
The results in this paper contribute to the literature on anomalies and the limits of arbitrage.Examples of related studies focusing on law of one price deviations in other markets include:closed-end funds (Lee et al. 1991), “negative stubs” or situations where the market value of acompany is less than its subsidiary (Mitchell et al. 2002), and American Depository Receiptsand cross-listed shares (Gagnon and Karolyi 2010) in equities; on-the-run versus off-the-run U.S. Treasuries (Krishnamurthy 2002), swap spreads (Duarte et al. 2007), mortgages(Gabaix et al. 2007), and Treasury-inflation-protected securities (Fleckenstein et al. 2014) infixed income; the CDS-bond basis (Garleanu and Pedersen (2011), Bai and Collin-Dufresne(2018)) in credit; and covered interest rate parity (Fong et al. (2010), Du et al. (2018))in foreign exchange. While these examples are only a subset from a large literature, theyserve to highlight how law of one price deviations occur across asset classes and are subjectto intense scrutiny given the model-free nature in which they challenge the conventionalwisdom. In some cases, the deviations are explained by frictions such as transaction costs,financing costs, or convergence risks. In other cases, the deviations are found to significantlypredict returns, posing a challenge for standard asset pricing models.
Within the limits-to-arbitrage literature, this paper most closely relates to studies ofequity volatility puzzles and anomalies. These studies can be broadly grouped into optionpricing and VIX futures papers. Beginning with the option pricing papers, two stylizedfacts since the crash of 1987 are that: (i) index options are expensive, with delta-hedged
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straddles earning a significant risk premium known as the variance risk premium, and (ii)out-of-the-money put options are particularly expensive, exhibiting a smirk or higher level ofimplied volatility relative to at-the-money options (Bates (2000), Coval and Shumway (2001),Jurek and Stafford (2015)). In response to these irregularities that are not explained bythe Black-Scholes-Merton model, the literature developed more sophisticated option pricingmodels that could account for some of the new patterns by allowing for stochastic volatilityand jumps in the underlying process (Heston (1993), Duffie et al. (2000), Madan et al.(1998)). But even with these improvements, the no-arbitrage models still struggle to fit andexplain some of the empirical properties of option prices (Bates 2003). Another strand of theliterature investigates the importance of demand-pressure and equilibrium effects (Grossmanand Zhou (1996), Bollen and Whaley (2004), Garleanu et al. (2009)). When dealers absorbdemand shocks from end-users or portfolio insurers, demand pressure can impact optionprices and the implied volatility surface, resolving some of the well-known option pricingpuzzles. An important conclusion from these studies is that demand pressure for one optionimpacts the prices of other options that have related, unhedgeable features.
Another puzzle from the equity volatility literature is that implied volatility shocks earn amuch smaller risk premium than realized volatility shocks (Dew-Becker et al. (2017), Andrieset al. (2015)). This result poses a challenge to consumption-based asset pricing models withEpstein-Zin preferences where investors would be willing to pay a premium for both typesof volatility shocks. In the data, the Sharpe ratios from receiving fixed in variance swapforwards and from selling VIX futures are much higher for shorter maturities, particularlyfor one-month variance swaps that are exposed to realized variance risk. Cheng (2018)builds on this result by studying the conditional price of implied volatility risk or the VIXpremium. The VIX premium is defined as the difference between the VIX futures priceand a statistical forecast of the VIX index at the maturity of the futures contract. This issimilar to the variance risk premium which is often defined as the VIX index, or a one-monthvariance swap rate, minus a forecast of realized variance. The VIX premium and variancerisk premium reflect the risk premiums for being exposed to implied volatility and realizedvolatility shocks respectively.
Cheng (2018) finds that the VIX premium declines when measures of systematic riskincrease. For example, the VIX premium declines when stock market returns are negativeor when realized or implied volatility increases. This result is surprising because the returnsfrom selling VIX futures are highly correlated with stock market returns. To the extent thatthe equity and VIX futures markets are integrated by a common stochastic discount factor,one would expect the VIX premium to increase along with the equity risk premium whenmeasures of systematic risk increase. In particular, Martin (2017) argues that the SVIX
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provides a lower bound for the equity risk premium, but the VIX premium is decreasing inmeasures of systematic risk like the SVIX, not increasing.
To partially resolve this puzzle, Cheng (2018) documents that dealer hedging demand forVIX futures declines when risk increases. One interpretation is that hedgers take profit onlong positions when risk increases, leading to the anomalous response of the VIX premiumto risk. Dong (2016) presents related work and argues that dealer positions in VIX futuresreflect hedging demand from VIX ETP issuance. Dong (2016)’s reduced form analysis andtheoretical model indicate that VIX ETP demand impacts the underlying VIX futures price.Mixon and Onur (2019) provide a related study of demand pressure using a detailed regula-tory dataset that includes dealer net positions by contract at a daily frequency, in contrast tothe public data used in other studies which includes weekly net position data that is aggre-gated across contracts. Mixon and Onur (2019) also find evidence of demand pressure withdemand increasing the average level and slope of the VIX futures curve. Similar to Garleanuet al. (2009), Mixon and Onur (2019) find that demand for one VIX futures contract spillsover to impact the prices of other contracts. However, the estimated demand effects arequantitatively small and typically within the spread between the no-arbitrage upper boundand lower bound for VIX futures.
This paper makes several contributions to the limits-to-arbitrage and equity volatilityliteratures. First, it provides a detailed, model-free measurement of no-arbitrage deviationsacross the VIX futures and S&P 500 index options markets. The no-arbitrage deviationmeasure is the difference between the price of a VIX futures contract and the price of a syn-thetic variance swap forward rate estimated from index option prices. Existing studies eitherestimate a risk premium for VIX futures that relies on time-series data and a parametricstatistical forecasting model as in (Cheng 2018) or they compute a deviation measure thatrelies on synthetic variance swap rates and a convexity adjustment from VIX options (Dong(2016), Park (2019)). This paper avoids using VIX options to focus instead on the syntheticvariance swap forward rates that are estimated from the larger and more liquid S&P 500index options market and because VIX options are only available starting in 2006.
Another advantage of the deviation measure in this paper is that it allows for a directand straightforward measurement of arbitrage opportunities. When the deviation measure ispositive, there is a static arbitrage opportunity because the price of a VIX futures contractis above its variance swap forward rate, its corresponding non-parametric upper bound.This paper provides new evidence on the frequency of no-arbitrage upper bound and lowerbound violations. The lower bound is estimated from the deviation measure and a convexityadjustment from a term-structure model. In addition, the paper shows that the deviationmeasure significantly predicts the returns of hedged VIX futures. This adds another predictor
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to the VIX futures literature to complement the VIX premium of Cheng (2018) and slopefactor of Johnson (2017) which also predict VIX futures returns. The predictability in thispaper is strongest when VIX futures are hedged with variance swap forwards, a relativevalue result that is new to the literature. Compared to existing studies, the size of thereturn predictability and Sharpe ratio in the trading strategy are large when VIX futuresare hedged with variance swap forwards.
Finally, this paper provides new evidence on the importance of the risk and demandchannels in driving the no-arbitrage deviations over time. In a VAR that accounts for theinteraction of risk and demand factors, the response to a risk shock is 3-4 times larger inmagnitude than the response to a demand shock over short horizons. The results indicatethat demand by itself is not sufficient to explain the negative response of the no-arbitragedeviation measure to risk. This observation coupled with the no-arbitrage violations docu-mented in the paper brings back into focus the puzzle as to why the prices of VIX futuresdecline relative to variance swap forward rates when risk increases.
3 Equity Volatility Markets
3.1 History and Products
Index options complete markets and expand the set of contingent claims that investors cantrade by allowing for the construction of Arrow-Debreu securities on the state of the stockmarket over different horizons (Ross (1976b), Breeden and Litzenberger (1978)). In practice,however, the option portfolios that span different payoffs can be complicated to constructand involve trading a significant number of options, resulting in large implementation costs.While narrower in focus, the variance swap and VIX futures markets allow investors to tradethe level of realized and implied volatility directly, without requiring a need for static ordynamic option trading strategies.
The S&P 500 index options started trading on the Chicago Board Options Exchange(CBOE) in 1983 but it was not until the late 1990s that the variance swap market startedto gain traction, perhaps encouraged by the high levels of index-option implied volatilitythat followed the Asian financial and LTCM crises (Carr and Lee 2009). The motivation fortrading variance swaps stemmed from investor demand to obtain exposure to the variancerisk premium and pure volatility risk. Although these payoffs can be approximated by adelta-hedged straddle, they cannot be spanned with a limited number of options. Consider adelta-hedged straddle. When the underlying moves away from the strike price, the exposureto realized and implied volatility changes. In Black-Scholes parlance, the gamma and vega
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of a delta-hedged straddle change with the moneyness of the option. In contrast, a varianceswap exchanges a fixed payment at maturity for a floating payment equal to the realizedvolatility of the underlying asset over the life of the swap, thus providing direct exposure torealized variance risk over different horizons.
The dawn of the variance swap market coincided with a number of theoretical develop-ments and influential papers that showed how to replicate the payoff of a variance swap witha static portfolio of options and a dynamic trading strategy in the underlying (Demeterfiet al. (1999), Carr and Madan (1999), Bakshi and Madan (2000), and Britten-Jones andNeuberger (2000)). This no-arbitrage replication argument is model-free in the sense thatit only requires the ability to trade a continuum of European options with different strikeprices, no jumps in the underlying asset, constant interest rates, and continuous trading ofthe underlying. Seeing the generality of this approach and the growth in the variance swapmarket, in 2003, the CBOE revised its definition of the VIX index to follow the no-arbitrageformula for pricing variance swaps with index option prices (Carr and Wu (2006), CBOE(2019)).
The CBOE then introduced trading in VIX futures and options in 2004 and 2006 respec-tively. The payoff to a VIX futures contract is the difference between the futures price anda special opening quotation of the VIX. From its definition, the VIX index upon which VIXfutures are based is equal to the square root of a one-month synthetic variance swap rate.The swap rate is “synthetic” because it is computed using the price of a portfolio of S&P 500index options. This relationship binds together VIX futures prices and index option pricesup to a convexity adjustment. The convexity adjustment is the definition of the VIX as thesquare-root of the price of the index option portfolio, not just the price of the portfolio itself.The square root convexity adjustment expresses the VIX index in volatility units that arefamiliar to option traders. The VIX is defined in the same unit as the Black-Scholes-Mertonimplied volatility parameter.
This paper relies on a detailed dataset of synthetic variance swap rates for the S&P 500index that are derived in Van Tassel (2019) to study the no-arbitrage relationship betweenthe index options and VIX futures markets. The synthetic variance swap rates are computedfrom the theoretical price for a variance swap following Carr and Wu (2009). The imple-mentation is similar to the computation of the VIX index but excludes the strike truncationand discretization that the CBOE uses. The paper then converts the synthetic varianceswap rates into variance swap forward rates for comparison to VIX futures prices. For ex-ample, a two-month variance swap can be decomposed into a one-month variance swap plusa one-month forward one-month variance swap. The ability to perform this decompositionis essential for the analysis in the paper. Variance swap forwards provide a non-parametric
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upper bound for the prices of VIX futures contracts (Carr and Wu 2006). This result servesas the basis for the main no-arbitrage deviation measure introduced in the paper.
In recent work, Martin (2017) has extended the generality of variance swap pricing. Byusing simple returns instead of log returns to compute the realized variance payoff for thefloating leg of a variance swap, Martin (2017) shows that the assumption of a continuousunderlying can be relaxed. Why does this paper use synthetic variance swap rates computedfrom the traditional formula rather than simple variance swaps instead? While simple vari-ance swaps have the advantage that they require no third-order approximation for jumps,they are not easily decomposed into forward rates and they do not serve as the basis for theVIX index. Thus, they do not have as close of a relationship to the pricing of VIX futures asdo synthetic variance swap rates and therefore are less well-suited for studying no-arbitragerelationships across the index options and VIX futures markets.
3.2 Market Size and Participants
To understand why no-arbitrage violations may occur across equity volatility markets, it ishelpful to provide a background on the institutional details of these markets. The relativesize of the different markets, types of participants, investor positioning, and linkages acrossmarkets are all potential determinants of no-arbitrage violations.
The S&P 500 index options and VIX futures markets are large and liquid exchange-basedmarkets. Figure 1 provides a brief summary of these markets including the growth in theiropen interest over time and a breakdown of investor positioning in VIX futures. The topleft plot illustrates the size of the index options market relative to the VIX futures marketover time. Both markets have exhibited significant growth over the past decade. S&P 500index options had an average open interest of $3.4 billion in Black-Scholes vega in 2018. VIXfutures had an average open interest of 462 thousand contracts in 2018, equal to $462 millionof “vega” or gains and losses for a one-point change in VIX futures prices given the contractmultiplier of $1000. While the VIX futures market experienced a nearly 10-fold increase inopen interest over the past decade, the index options market is still 7-times larger in 2018as measured by vega. To the extent that larger market size corresponds to greater liquidityand more informative prices, the index option market provides a valuable signal about thefair value of VIX futures prices. The no-arbitrage deviation measure introduced later in thepaper is motivated by this idea, using index option prices to bound VIX futures prices.
Focusing on the VIX futures market, the top right plot shows that post-crisis growthcoincides with the rise of volatility-exchange-traded products (VIX ETPs). VIX ETPs issuedby banks and broker-dealers allow retail investors to gain exposure to implied volatility
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without needing to trade in the index options or VIX futures markets directly. For example,consider the VXX, one of the first ETPs introduced to the market in early 2009. TheVXX ETN tracks the S&P 500 VIX Short-Term Futures Index, rolling from the front-month contract to the second-month contract to provide investors with a weighted averagefutures maturity of one-month. Instead of needing to perform the roll themselves, investorscan simply purchase and hold the exchange-traded note. Over time, similar products wereintroduced that track different VIX futures indices with long or short exposure, potentiallybuilding in embedded leverage. The top right plot of figure 1 illustrates that the growth ofVIX futures has coincided with the growth of VIX ETPs.
One explanation for the simultaneous growth in the VIX futures and ETP markets isthat dealers issue VIX ETPs to satisfy retail demand and then hedge in the underlying VIXfutures market. Dong (2016) suggests this hypothesis and shows that short-term ETP netvega is highly correlated with dealer positions from the CFTC’s Commitment of Traders(CoT) Report.1 The bottom left plot replicates and extends this result. Dealer positionsclosely match the magnitude of VIX short-term ETP net vega and the two demand variablestrack each other with a correlation of around 60% from 2010 to 2018. This result suggests thatretail demand may be an important determinant for the growth in the VIX futures market.If retail demand to buy volatility causes dealers to hedge in the VIX futures market, thiscould affect VIX futures prices and drive them above the no-arbitrage bounds implied bythe index option market.
If dealer positions in the VIX futures market are just a veil for retail demand, whoabsorbs the retail demand shocks? The bottom right plot attempts to address this questionby plotting net positions for different trader types including dealers, leveraged funds, assetmanagers, and other reportable traders from the CFTC’s CoT Report. Focusing on thepost-crisis period, the largest net position by magnitude belong to dealers and leveragedfunds. Dealer and leveraged funds net positions exhibit a correlation of -91% from 2010 to2018. One interpretation of this result is that leveraged funds absorb retail demand shocksthat are passed through by dealers from the volatility ETP market.
With this context in mind, the paper next describes the no-arbitrage bounds on VIXfutures prices and introduces a deviation measure which tracks arbitrage opportunities over
1The VIX ETP demand variable is equal to the leverage-weighted market capitalization of short-termETPs net of short interest. Following Dong (2016), the formula is DETP =
∑i∈ST (Shrouti − ShortInti) ·
Pi ·Mi where Shrouti is shares outstanding, ShortInti is short interest, Pi is price, and Mi is the directionand leverage multiplier. I compute DETP using Bloomberg data for the VXX, VIIX, VIXY, UVXY, TVIX,XIV, SVXY, IVOP, XXV, and VXXB ETPs. The multipliers are equal to M = [1 1 1 2 2 -1 -1 -1 -1 1]. Iadjust the multipliers for UVXY and SVXY to 1.5 and -.5 after 2/28/18. To express this demand variable asVIX Short-Term ETP Net Vega in $ million of vega in the plot, I divide DETP by the VIX index. Similarly,the total vega number is
∑i∈ST Shrouti · Pi · |Mi|/V IX.
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time and across contracts. The paper finds that there are arbitrage opportunities across theVIX futures and index option markets. The deviation measure widens when risk increasesand predicts VIX futures returns. The paper then returns to the demand variables describedabove to investigate the relationship between the no-arbitrage deviation measure, risk, anddemand pressure.
4 No-Arbitrage Deviation Measure
4.1 VIX Futures Bounds
The upper bound for the price of a VIX futures contract is the maturity-matched one-monthvariance swap forward rate expressed in volatility units. The derivation of the upper boundfollows from the definition of the VIX index and an application of Jensen’s inequality (Carrand Wu 2006). The VIX is defined as the square-root of a one-month variance swap rate. Bytaking the square-root, the definition expresses the VIX in volatility units that are familiarto traders who use the Black-Scholes-Merton model. At the same time, the square-rootintroduces a wedge between the price of VIX futures and their corresponding variance swapforward rates.
To derive the upper bound, let the price of a VIX futures contract be the risk-neutral Qexpected value of the VIX index at maturity,
Futt,T = EQt [V IXT ]. (1)
As noted above, the VIX is defined as the square-root of a one-month variance swap rateV IXt ≡
√V St,t+1.2 Let variance swaps be modeled as the risk-neutral expected value of
realized variance from the trade date until the maturity of the swap,
V St,T = EQt [RVt,T ] . (2)
Similarly, let variance swap forward rates be the risk-neutral expected value of realizedvariance between two forward starting dates,
Fwdt,T1,T2 = EQt [RVT1,T2 ] . (3)
2In practice the VIX index is expressed in annualized volatility units. The paper omits the annualizationin this derivation for notional simplicity. Empirically the paper compares VIX futures prices to varianceswap forward rates in annualized units, using calendar time to perform the annualization as in the CBOEconstruction of the VIX (CBOE 2019).
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It follows that the prices of VIX futures are bounded above by variance swap forward rates,
Futt,T = EQt [V IXT ]
= EQt
[√V ST,T+1
]≤
√EQ
t [V ST,T+1]
=√EQ
t
[EQ
T [RVT,T+1]]
=√Fwdt,T,T+1.
(4)
The one-month variance swap forward rate from time T to time T + 1 expressed in volatilityunits is
√Fwdt,T,T+1.
If VIX futures violate the upper bound so that Futt,T >√Fwdt,T,T+1, there is a static
arbitrage opportunity . To see this, note that V ST,T+1 = f(V IXT ) = V IX2T is a convex
function. This motivates the trade:
Arbitrage Trade for Upper Bound Violation
Time t T
Sell f ′(Futt,T ) VIX futures contracts 0 f ′(Futt,T )(Futt,T − V IXT )
Pay fixed in VS forward 0 V ST,T+1 − Fwdt,T,T+1
Total 0V ST,T+1 − f(Futt,T )− f ′(Futt,T )(V IXT − Futt,T )
+f(Futt,T )− Fwdt,T,T+1
The first term in the payoff at time T is non-negative because f is convex. The secondterm is positive because f(Futt,T ) = Fut2t,T > Fwdt,T,T+1 by assumption. The trade is anarbitrage because it requires no investment at time t and guarantees a positive payoff attime T .
The prices of VIX futures contracts are also bounded below by volatility swap rates. Thederivation of the lower bound proceeds along similar lines. In particular, the price of a VIXfutures contract satisfies,
Futt,T = EQt
[√EQ
T [RVT,T+1]
]≥ EQ
t
[EQ
T
[√RVT,T+1
]]= EQ
t
[√RVT,T+1
]≡ Fvolt,T,T+1,
(5)
where the last line defines a volatility swap forward rate. In the discussion below the paperwill focus on the upper bound because the upper bound can be estimated non-parametricallydirectly from observed variance swap rates. To compute the lower bound the paper will rely
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on a dynamic term-structure model to estimate the difference between the upper and lowerbounds following the approach in Van Tassel (2019).
4.2 No-Arbitrage Deviation Measure
The ability to bound VIX futures prices with variance swap forward rates motivates definingthe no-arbitrage deviation measure,
Deviationt,n ≡ Futt,n − Fwdt,n. (6)
This deviation measure is the difference between the price of the n-th futures contract Futt,nexpiring at time τn and the corresponding one-month variance swap forward rate Fwdt,n
expressed in volatility units. The deviation measure is defined in levels rather than as apercentage because the absolute size of the no-arbitrage violations matter. Abstracting fromtransaction costs, arbitrageurs should trade against a large upper bound violation, regardlessof whether the VIX is at 20 or 50. Note also the slight abuse of notation compared to theprevious section. In the deviation measure, Fwdt,n denotes the variance swap forward ratein volatility units to be directly comparable to VIX futures prices.
The deviation measure is attractive because of its straightforward interpretation andfor its ability to be estimated non-parametrically with minimal assumptions. When thedeviation measure is positive, VIX futures prices are above their upper bound which createsa static arbitrage opportunity or law of one price violation. When the deviation measureis negative, VIX futures are cheap relative to variance swap forwards to the extent thatthe bound is relatively tight. Large negative deviations may also violate the lower bound,but this is harder to measure because the prices of volatility swaps are harder to observe.Compared to variance swaps, volatility swaps are traded infrequently and are difficult toreplicate from index option prices. Therefore, the paper uses the upper bound from varianceswap forward rates to define the deviation measure.
Computing the deviation measure requires the prices of VIX futures and of variance swapforwards. The VIX futures price is directly observable from settlement prices or intradaytrade-and-quote data. The variance swap forward price can be computed directly froma variance swap curve. The paper computes the variance swap forward curve assumingflat forward rates between observed variance swap maturities.3 Figures 2 and 3 provide two
3For example, let V St,T1and V St,T2
be variance swap rates at maturities T1 and T2. The forwardrates between T1 and T2 are assumed to be constant and equal to Fwdt,T1,T2
in annualized variance unitswhere Fwdt,T1,T2
satisfies V St,T1(T1 − t) + Fwdt,T1,T2
(T2 − T1) = V St,T2(T2 − t). Let F (τ) denote the
corresponding forward curve, i.e. F (t, τ) = Fwdt,T1,T2 ∀τ ∈ (T1 − t, T2 − t) and a similar definition for othermaturities τ . The upper bound for the n-month VIX futures contract expiring at time τn is then equal to
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examples of this computation on two dates with large no-arbitrage deviations. Figure 2 plotsthe variance swap curve, instantaneous forward curve, and one-month forward rates for theVIX futures maturity dates. Figure 3 compares the VIX futures prices to their correspondingvariance swap forward rates. Table 1 provides the corresponding data, including the syntheticvariance swap curve from which forward rates are estimated and the VIX futures prices. Theexample on February 27, 2012 illustrates a case when the prices of VIX futures are abovethe upper bound resulting in a static arbitrage opportunity. The size of the violation islarge as the deviation measure is more than 1% for several of the contracts. The example onAugust 10, 2011 illustrates the opposite case when the prices of VIX futures are well belowthe upper bound, often by as much as 3% to 5%. Using an estimate from a term-structuremodel for the difference between the upper and lower bounds for the different contracts onthis date, it appears that VIX futures prices are below the lower bound. This indicates anarbitrage opportunity if arbitrageurs are able to transact at volatility swap forward ratesnear the lower bound estimates.
The paper computes the deviation measure with synthetic variance swap rates fromVan Tassel (2019) and synchronized VIX futures prices. The synthetic variance swap ratesare constructed from index option prices from OptionMetrics data for maturities of τ = {1,2, 3, 6, 9, 12, 15, 18, 24} months. To be synchronous with the option quotes, the VIX futuresprices are settlement prices from 4:15pm ET prior to March 3, 2008 and then mid-prices at4pm ET from Thomson Reuters Tick History when available. This change in timing matchesthe OptionMetrics quotes which switched to 4pm in March 2008. As a robustness check,the Appendix computes the deviation measure in a number of alternative ways and findsresults that are broadly the same. The baseline deviation measure is 97% to 100% correlatedwith alternative measures that make a convexity adjustment to reduce the bias in the upperbound, either using a term-structure model or regression. In addition, the baseline measureis around 90% correlated with deviation measures computed from Bloomberg and CBOEvolatility index data and 70% to 80% correlated with deviation measures computed fromOTC variance swap quotes. The baseline measure has the advantage that it is available overthe full sample period from March 2004 to December 2018, whereas the alternative datasetsare only available for part of the sample period.
Fwdt,n =√∫ τn+1
τnF (t, s)ds. Similar qualitative results are obtained if the flat forward rates are smoothed
with a moving average filter like a lowess smoother (unreported in the paper).
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4.3 Deviation Measure Time-Series
The deviation measure varies significantly over time. Figure 4 plots the average deviation andaverage absolute deviation for the front six contracts from March 2004 to December 2018.During the first part of the sample period the average deviation exhibits positive valueson a significant fraction of dates indicating the presence of arbitrage opportunities. Since2012 the average deviation declines from around -.50% to -1.00% and is positive less often.The plot also highlights how the deviation measure responds to changes in risk. Aroundnegative events that coincide with stock market declines like the financial crisis, equity flashcrash, and S&P downgrade of U.S. debt, the deviation measure decreases while the absolutedeviation increases in magnitude.
Table 2 presents summary statistics of the deviation measure for the front six contractsfrom 2007 to 2018 when a balanced panel is available. The last column averages the statisticsacross contracts. The results indicate that the deviation measure is negative on averagewith a bias that increases from around -.25% for the front contract to around -1% for thelonger-dated contracts. The negative bias is statistically significant as indicated by the t-statistics and consistent with the definition of the deviation measure which equals the VIXfutures price minus an upper bound. In terms of variability, the deviation measure hasa standard deviation of around .80% across contracts and exhibits negative skewness andexcess kurtosis, particularly for the first and second contract. The deviation measure is alsopositively autocorrelated across one-day, one-week, and one-month horizons. The results areconsistent with the time-series plot, indicating that there are large and persistent law of oneprice deviations across the VIX futures and index options markets.
The summary statistics also highlight differences across contracts. Panel B reports thecorrelation matrix of the deviation measure. While the pairwise correlation is positive acrosscontracts, it tends to decline as the contract maturities get further apart and is lowest forthe front contract. The average pairwise correlation for the other contracts is around 50%.Panel C reports how the average deviation varies across contracts over time. The resultsindicate that the decline in the average deviation from the time-series plot is driven by thelonger-dated contracts rather than the front contract. Combined with the previous results,the summary statistics indicate that the deviation measure exhibits substantial variabilityboth over time and within the cross-section.
4.4 The Frequency of Law of One Price Violations
Table 3 reports the frequency of law of one price violations. Panel A.I reports the fractionof days when the deviation measure exceeds a positive threshold Deviationt,n ≥ τ where τ ∈
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{0, .25, .50} over the full 2004 to 2018 sample period. Positive values of the deviation measureindicate the presence of arbitrage opportunities in which the prices of VIX futures are abovetheir upper bound from variance swap forward rates. The table indicates that almost 20%of contract-date observations correspond to upper bound violations. The violations are mostfrequent for the front and second contracts which exhibit positive values for Deviationt,n on35% and 24% of days respectively. The results for the lower bound violations in Panel B.Iis similar. Nearly 20% of contract-date observations correspond to a lower-bound violationwith these violations occurring more frequently for longer-dated contracts. Taken together,the results indicate that VIX futures exhibit frequent law of one price violations relative totheir no-arbitrage bounds.
Are these results driven by the early years in the sample, small upper bound violations,or longer-dated less liquid contracts? The answer is no. Panels A.I and B.I show thatupper bound and lower bound violations occur for all contracts. Panels A.II and B.II reportthe number of upper bound and lower bound violations from 2010 to 2018. For the frontcontract, the number of upper bound violations remains around 35% in the post-crisis period.The average number of upper bound violations declines from 19% to 12% of contract-dateobservations, but this remains a substantial fraction of violations. Moreover, the number oflower bound violations increases from 22% to 30%, with large increases for the longer-datedcontracts. This rules out the hypothesis that the arbitrage violations are solely driven by theearly years in the sample; these violations do not go away after liquidity improved or aftertraders learned of the no-arbitrage relationships. To illustrate this point graphically, Figure5 reports the deviation measure for the front contract and the VIX futures price and varianceswap forward rate. The top plot shows that the deviation measure indicates consistent lawof one price violations across the entire sample period. The bottom plot shows that the VIXfutures price and variance swap forward rate track each other closely throughout the entiresample period. Thus, it does not appear that the violations are driven by the early years inthe sample. Instead, they are pervasive.
The no-arbitrage violations are also large in magnitude. The table indicates that around10% (5%) of contract-date observations have upper bound violations greater than .25%(.50%) over the full sample period. For the lower bound there are around 14% (8%) ofcontract-date observations with violations lower than -.25% (-.50%). The change in the2010-2018 period for the large violations is similar to the change for the baseline violationsaround τ = 0. These results also address concern about measurement noise. In particular,estimation of variance swap forward rates introduces some noise in the deviation measurewhich is a potential concern for measuring the frequency of arbitrage opportunities. The factthat a significant fraction of law of one price violations are of large magnitude assuages this
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concern. In addition, the time-series plots show a clear relationship between the estimateddeviation measure and risk events. If measurement noise dominated the results, these plotswould appear much noisier and the deviation measure would not predict the returns of VIXfutures. The next section investigates return predictability and finds that the deviationmeasure is highly significant at predicting returns, consistent with the idea that it identifiesno-arbitrage violations between the prices of VIX futures and variance swap rates.
5 VIX Futures Return Predictability
5.1 VIX Futures and Variance Swap Forward Returns
To study return predictability, the paper defines the excess return from selling the n-th VIXfutures contract over horizon h as,
RV IXt+h,n = Futt,n − Futt+h,n. (7)
Similarly, the paper defines the excess return from receiving fixed in a one-month varianceswap forward with a maturity matched to the n-th futures contract as,
RV SFt+h,n = Fwdt,n − Fwdt+h,n, (8)
where the forward price Ft,n is expressed in variance units. The Appendix shows that similarqualitative results hold for log- and percentage-returns.4 While percentage-returns maybe preferred in other settings, such as for stocks where prices are not stationary and thepercentage-return is the return on investment from buying one share, this is not the casefor VIX futures. The return defined above RV IX
t+h,n is the return an investor would receivefrom selling one contract, abstracting from the multiplier and margin for simplicity. Incontrast, generating percentage-returns for VIX futures requires a more complicated tradingstrategy that adjusts the position size each day to account for the level of the futures price,a difference that would entail additional transaction costs in practice. The trading strategydiscussed below will define returns that are similar to RV IX
t+h,n taking the multiplier, margin,and transaction costs into account.
Table 4 reports summary statistics for VIX futures and variance swap forwards pricesand returns. The table includes the front six contracts from 2007 to 2018 when a balancedpanel is available and reports the average statistic across contracts in the last column. For
4The log- and percentage-returns are defined as Futt+h,n/Futt,n − 1 and log(Futt+h,n/Futt,n).
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prices, the table indicates that the unconditional term-structure is upward sloping. Thestandard deviation is decreasing in the contract number reflecting the mean-reversion ofimplied volatility. The prices are positively skewed and exhibit excess kurtosis. For returns,the average return tends to be positive but is not statistically significant. The exception isthe fifth and sixth contract for variance swap forwards where the average return is actuallynegative. These results are consistent with Dew-Becker et al. (2017) and Andries et al. (2015)who find a large risk premium from being exposed to realized volatility, but a smaller riskpremium for being exposed to implied volatility. The returns also exhibit negative skewnessand excess kurtosis which is common for volatility selling strategies.
5.2 Return Predictability
Table 5 reports the main return predictability regressions. The regression is a two-stepprocedure. First, VIX futures are regressed onto variance swap forward returns to obtain ahedge ratio for each contract βn. The variance swap forward returns are highly significantfor VIX futures returns, exhibiting an average explanatory power of around 70% to 80% asreported in the Appendix. The hedged returns are then regressed onto the deviation measureto see whether the no-arbitrage deviation predicts VIX futures returns that are hedged withvariance swap forward returns. The idea is that, when the deviation measure is high, VIXfutures are expensive relative to variance swap forwards, so the returns from selling VIXfutures and paying fixed in variance swap forward rates should be high. Similarly, whendeviation is low, VIX futures are inexpensive relative to variance swap forward rates so thereturns from selling VIX futures should also be low. The table confirms this hypothesiswith the highly significant and positive point estimates on the deviation measure acrossspecifications.
Panel A reports results for the full sample from 2004 to 2018 at a weekly horizon h = 5.For ease of interpretation, the dependent and independent variables are standardized. Forexample, in Panel A.I, a one-standard deviation increase in the deviation measure for thesecond contract predicts a .23 standard deviation higher return with an R2 = 5.1%. A one-standard deviation increase in the deviation measure for the sixth contract predicts a .38standard deviation higher return with an R2 = 14.5%. The average R2 and t-statistic acrosscontracts are 8.3% and 5.7 respectively. Panel A.II shows that these results are robust toincluding the VIX, realized variance over the past 21 days (RV ), the CRSP value-weightedstock market return over the past week (RMRF ), and volume for the n-th contract normal-ized by open interest (V LM). If anything, including these additional variables increases thestrength of the deviation measure as a predictor and also increases the in-sample explanatory
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power.Are these results specific to the sample period, investment horizon, or deviation measure
specification? The answer is no. The deviation measure robustly predicts the returns ofVIX futures hedged with variance swap forward rates across the specifications explored inthe paper and Appendix. For example, Panel B runs the regressions for a post-crisis sampleperiod from 2010 to 2018 and finds similar results. If anything, the predictability in thepost-crisis period increases for the second contract but remains stable for the other contracts.The average R2 and t-statistic for the deviation measure in Panel B.II are 12% and 7.52.The Appendix reports additional results showing that the predictability holds over dailyand monthly horizons, for lagged and bias-adjusted versions of the deviation measure, andwhen the deviation measure is computed with synthetic variance swap rates from alternativedatasets like Bloomberg data or the CBOE volatility indices.
The evidence suggests that the deviation measure significantly predicts the returns ofVIX futures hedged with variance swap forwards. As noted before, these results providereassurance that the deviation measure is picking up a no-arbitrage deviation and does notmerely reflect measurement noise from estimating variance swap forward rates. But if thedeviation measure is picking up mispricing, is the VIX futures or index option market drivingthe predictability? Since the analysis is relative, it is challenging to provide a sharp answer tothis question. One hypothesis is that the larger and more established index options marketwould serve as a fair-value measure for VIX futures. Alternatively, given the high liquidityof VIX futures in the latter part of the sample and the fact that VIX futures contractsare traded directly, rather than being derived from option portfolios across maturities, onemight hypothesize that VIX futures would be fairly priced with variance swap forward ratesdriving the predictability. One way to test these hypotheses is through the predictabilityof the deviation measure when the returns of VIX futures and variance swap forwards arehedged with stock market returns. If the deviation measure is picking up mispricing foreither VIX futures or variance swap forwards, then it should remain significant at predictingthe returns in these markets after hedging with returns from another market, namely thestock market.
Table 6 reports the return predictability results for VIX futures hedged with CRSP value-weighted returns. As before, VIX futures returns are first regressed onto stock market returnsand then the hedged returns are regressed onto the deviation measure. The explanatorypower of the stock market is somewhat weaker for VIX futures returns than using varianceswap forward returns for hedging as in the previous table (see the Appendix). Despite this,the results indicate that the deviation measure is still significant at predicting VIX futuresreturns hedged with the stock market. While the predictive power is somewhat lower, in
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Panel A, the point estimates are all positive and significant, except for the third contractwhich is only positive. Similar to the previous results, Panel A.II shows that the results arerelatively stable after including additional predictors like the VIX and realized variance toproxy for the variance risk premium. During the post-crisis period, Panel B indicates thatthe point estimates remain significant or become more significant in the case of the thirdcontract. Interpreting these results, while hedging VIX futures with variance swap forwardsfrom the previous table may be closer to a textbook arbitrage, the results from Table 6indicate that the deviation measure still exhibits some predictability for VIX futures evenwhen they are hedged with the stock market.
Table 7 then performs the analogous regressions for variance swap forwards hedged withCRSP value-weighted returns. In this case, the expected sign on the deviation measureis negative to the extent that the deviation measure picks up mispricing in variance swapforwards relative to VIX futures. In Panel A, there is some evidence of predictability forvariance swap forwards matched to contracts three to six, but no evidence for contracts oneand two which have the wrong sign. In Panel B.II, deviation predicts hedged variance swapforward returns with maturities matched to the fifth and sixth contract, but not for contractsone through four. The average R2 and t-statistic in Panel B.II across contracts are 2.2%and -1.68 which contrasts 4% and 3.4 in Table 6 when VIX futures are hedged with thestock market. While there is some evidence of predictability for variance swap forward rates,compared to the previous table, the results suggest that the deviation measure is picking uplarger amounts of predictability for VIX futures than for variance swap forward rates.
5.3 Relative Value Trading Strategy
The results from the previous section indicate that the no-arbitrage deviation measure sig-nificantly predicts the returns of VIX futures hedged with variance swap forwards acrossvarious in-sample specifications. How robust is this predictability over time and how can itsmagnitude be interpreted? Does accounting for financing or transaction costs eliminate thepositive CAPM alpha?
One way to investigate these questions is with a trading strategy. To that end, the papercomputes the returns from a basic strategy that sells (buys) VIX futures contracts when thedeviation measure exceeds a high (low) threshold value and hedges with either variance swapforwards or stock market returns. To normalize the deviation measure within contract andover time, the strategy converts the deviation measure into a rolling z-score that is computed
20
using one-year of lagged data,
Zt,n ≡Deviationt,n − µt,n
σt,n. (9)
Similarly, the hedge ratio for each contract βt,n is computed from rolling regressions with one-year of lagged data. The regressions are analogous to the first step in the return predictabilityregression from the previous section, except that they use rolling instead of full sample data.This approach accounts for any time-variation in the hedge ratios and ensures that the hedgeratios are in the investor’s information set.
Returns for the baseline trading strategy for a short position in the n-th VIX futurescontract are defined as,
Rt+h,n ≡1000 · (Futt,n − Futt+h,n − βhedge
t,n ·Rhedget+h,n)−Margint,n ·Rft · h
365
Margint,n
. (10)
This definition uses maximal leverage to margin following Garleanu and Pedersen (2011).The payoff is the change in the VIX futures price minus the hedging return times the contractmultiplier. The margin is the initial margin for the n-th VIX futures contract. The financingcost is the margin multiplied by the risk-free rate times the actual number of days h over365 days in a year. The proxy for the risk-free rate is the three-month U.S. Treasury billrate. The return for a long position is defined analogously.
One aspect of the return definition is that it abstracts from the financing cost for thehedging return. On one hand, this assumption may be conservative. If an exchange orbilateral counterparty offered portfolio margin, the capital requirement for the hedged tradewould be lower than the margin for a naked position in VIX futures because of the reducedrisk due to the hedge. On the other hand, if an investor was subject to a leverage constraintor if margin was required for both the VIX futures trade and hedge, more capital might berequired. The paper focuses on the margin for VIX futures contracts because that data isavailable historically. It is less clear, for example, what the margin would be on a bilateral,over-the-counter variance swap forward trade in the pre-crisis or post-crisis era. While someof the results are sensitive to this margin assumption, most are not. For example, the CAPMα-to-margin is determined in part by the maximum leverage, which is directly related to themargin assumption. In contrast, doubling or tripling the margin would have no impact onthe Sharpe ratio of the strategy.
The return predictability regressions indicate that the deviation measure significantlypredicts the returns of VIX futures hedged with variance swap forwards across the front sixcontracts. Motivated by this, the paper focuses on a baseline trading strategy that hedges
21
with variance swap forwards, has a one-week horizon h = 5, and uses a threshold value ofτ = .50 z-scores to trade the front six contracts. If there are multiple contracts to trade onthe same day, the strategy forms an equal-weighted portfolio. As a robustness check, thepaper also explores alternative strategies that vary the threshold value, number of contractstraded, and hedging instrument.
Figure 6 plots the performance of the relative value strategy in comparison to the stockmarket. The returns are normalized to 10% annualized volatility for comparison. From 2004to 2018 the VIX futures trading strategy earns a Sharpe Ratio (SR) of 3.0 versus .5 forCRSP value-weighted returns. The plot reports the cumulative sum of overlapping weeklyreturns for both series. The VIX futures strategy performs well across market environmentswith minimal drawdowns compared to the stock market. Even during the financial crisis,the VIX futures strategy continues to earn positive returns.
Table 8 builds on these results by reporting summary statistics for the trading strategyreturns compared to stock market and volatility factor returns. Panel A reports summarystatistics for the relative value trading strategy across specifications and time periods. PanelB reports summary statistics for stock market and volatility factor returns. The first columnin Panel A corresponds to the performance of the baseline strategy from Figure 6 that tradesthe front six contracts with a threshold of τ = .50 and hedges with variance swap forwards(VSF). The strategy returns are adjusted to have 1.39% weekly volatility for
√52× 1.39%
= 10% annualized volatility as in the time-series plot. The weekly SR of .42 annualizes to√
52× .42 = 3.0 matching Figure 6. Beyond this metric, the strategy delivers positivelyskewed, fat-tailed returns that are only negative 27% of the time. The maximum drawdownof 6.7% contrasts a maximum drawdown of around 50% for the stock market over the sameperiod of time (Panel B, column 1). To obtain 10% annualized volatility, the returns arede-levered and multiplied by 11.6%. The unlevered mean return is 4.99%. This is close tothe alpha-to-margin of 4.90% per week, indicating that the maximally leveraged strategydelivers large returns that are largely unexplained by the market factor.
The subsequent columns in Table 8 vary aspects of the trading strategy or sample periodto examine the robustness of the results. The second column (2) in Panel A uses stockmarket returns (RMRF ) as a hedge instead of variance swap forward returns. Similar tothe return predictability regressions, the performance declines somewhat but the strategystill obtains an annualized SR of 1.43 and CAPM alpha-to-margin of 3.6%. The returnsremain positively skewed and continue to exhibit a small maximum drawdown of 10%.
The third column (3) in Panel A adds transaction costs to the strategy from column(2) that hedges with the stock market. The returns with transaction costs incorporate thebid-ask spread from closing quote data for VIX futures. For example, when selling VIX
22
futures, the strategy sells at the bid and buys the contract back at the ask a week later. Thestrategy does not trade if the bid-ask spread exceeds .25 volatility units, which is five timesthe typical bid-ask spread in recent years. This constraint is included to avoid trading duringilliquid periods of time when it may be costly or difficult to execute the strategy. As willbe shown below, the strategy with transaction costs performs better when trading the fronttwo contracts instead of the front six contracts. The front two contracts are more liquid andhave lower transaction costs on average. Using the same threshold of τ = .50 as before, thestrategy with transaction costs and a stock market hedge earns an annualized SR of .73 witha CAPM alpha-to-margin of 2.6% per week. The returns continue to be positively skewedand are only negative 32% of the time. The next columns (4) to (6) show that similar resultshold in a post-crisis sample from 2010 to 2018. The SRs and CAPM alpha-to-margin areslightly lower, but still large. The maximum drawdown and percentage of negative returnsare similar to the full sample estimates.
Panel B presents summary statistics for stock market and volatility returns for compar-ison. Column (1) reports results for CRSP value-weighted returns (RMRF). Columns (2)and (3) report results for receiving fixed in one-month variance swaps and selling the front-month VIX futures contracts. The horizon is one-week and the returns are computed overthe same period of time as for the trading strategy returns in Panel A. The annualized SRsfor the stock market, variance swap, and VIX futures returns are .51, 1.08, and .65. Whilethe relative value strategy earns a higher SR than the stock market and the strategy ofunconditionally selling front month VIX futures, the SR from receiving fixed in one-monthvariance swaps is similar to the relative value strategy when hedging with stock market re-turns. Unlike the relative value strategy, the stock market and volatility selling returns arenegatively skewed. The stock market returns and returns from unconditionally selling thefront-month VIX futures contracts also exhibit large drawdowns during the financial crisis.
The summary statistics indicate that the trading strategy earns excess returns relative tothe CAPM as measured by the large alpha-to-margin estimates. How significant is this theoutperformance and is it robust to other factor models? Table 9 answers this question byreporting alpha-to-margin estimates for different specifications including the CAPM, a four-factor model with the Fama-French three factors and Carhart momentum factor (FFC4), anda six-factor model (FFCV6) that adds realized and implied volatility factor returns fromone-month variance swap forwards (VS1) and front-month VIX futures contracts (VX1).Columns 1-3 report results for the full sample period and columns 4-6 report results for apost-crisis sample from 2010 to 2018.
The baseline strategy that hedges with variance swap forwards earns an alpha-to-marginof 4% to 5% across specifications and sample periods. The insignificant point estimates on
23
the factors and low R2s indicate that the variance swap forward hedge has removed almost allof the systematic risk in the trading strategy returns. The large alpha and low explanatorypower of the factor models is consistent with the idea of trading against an arbitrage spread.Panels B and C find similar qualitative results for the other trading strategy specifications.The alpha estimates remain significant and the explanatory power of the factor modelsremains low across specifications. While the volatility factors are more significant in PanelsB and C than in Panel A, the highest R2s are only 10% to 15% and the negative loadingson the volatility factors generally increase the alpha estimates. Overall the alpha estimatesappear robust to different factor model specifications and time periods.
The top plot in Figure 7 reports the performance of the three main specifications thathedge with variance swap forward rates, the stock market, and the stock market with trans-action costs. The relative value strategy hedged with variance swap forwards or stock marketreturns outperforms the stock market over nearly the entire sample period. The strategywith transaction costs underperforms until the financial crisis and then starts to outperformthe stock market. The higher values for the relative value strategies at the end of the samplerelative to the stock market are consistent with their higher SRs from the summary statisticstable. Note also that the stock market is a difficult benchmark to outperform during thissample period. Alternative equity risk factors such as value (HML), small-minus-big (SMB),and momentum (MOM) generally underperformed the stock market, earning annualized SRsof .01, .08, and .12, that also underperform the relative value strategy.
Is the outperformance of the relative value strategies driven by the trading threshold ornumber of contracts traded? The bottom plot in Figure 7 investigates this question. Thereare two takeaways for the baseline strategy that hedges with variance swap forward rates.First, as the number of contracts traded increases, the SR increases. This result is consistentwith the return predictability regressions. Since the deviation measure predicts returnsacross all contracts, trading more contracts provides the strategy with more relative valueopportunities to exploit, improving performance. For example, the SRs with a threshold ofτ = .50 when trading up to the front 1, 2, 4, and 6 contracts are .92, 1.58, 2.63, and 3.00.
The second observation is that the SRs tend to peak at an intermediate trading thresholdτ and then decline. For large thresholds the strategy trades less frequently and only againstlarge deviations, passing up valuable trading opportunities. This pattern is slightly different,however, when including transaction costs. In this case, it is still desirable to trade againstlarge opportunities, but there is also a benefit to trading less frequently to lower transactioncosts. This interpretation echoes findings from the literature on optimal portfolio choicewith transaction costs (Davis and Norman 1990; Gârleanu and Pedersen 2013). When in-corporating transaction costs, the SRs do not decline as rapidly with the trading threshold
24
and in some cases it is better to use a larger threshold. The SRs are also higher when onlytrading the front 1 or 1-2 contracts, reflecting the lower transaction costs of these contractsrelative to the longer-dated contracts. The Appendix shows that similar results hold in the2010 to 2018 post-crisis period.
In summary, the results indicate that trading against the no-arbitrage deviation measureearns significant returns with minimal exposure to traditional risk factors. The alpha-to-margin estimates and high SRs are robust across a range of specifications and sample periods.The strongest results correspond to the strategy that hedges with variance swap forwards.This strategy is most similar to the textbook arbitrage trades between VIX futures andvariance swap forwards or volatility swap forwards described earlier in the paper.
6 Discussion
6.1 No-Arbitrage Deviation versus Risk and Demand Factors
What drives the no-arbitrage violations and deviation measure over time? Two channelsidentified by the limits-to-arbitrage literature are risk and demand. If arbitrageurs are riskaverse or have limited capital, demand shocks can push prices away from fundamental values.The resulting effect of limits-to-arbitrage and demand shocks can be anomalies such as returnpredictability and no-arbitrage violations.
The paper estimates a vector autoregression (VAR) to study how the no-arbitrage devia-tion measure relates to risk and demand factors, yt = [DEVt V IXtDNPt]. The variables inthe VAR are the average deviation measure across the front six contracts (DEV), the CBOEVolatility index (VIX), and the dealer net position (DNP) in VIX futures from the CFTC’sCommitment of Traders Report (CoT). The DEV variable focuses on the average deviationto keep the size of the VAR small. The VIX and DNP variables serve as proxies for riskand demand. Figure 1 motivates using DNP as a demand variable by illustrating the highcorrelation between DNP and VIX ETP demand, a proxy for retail demand. The advantageof DEV relative to VIX ETP demand is that it only depends on quantities, not on prices.The DNP variable in the VAR is the dealer net position as a fraction of open interest. Thisnormalization bounds DNP between 0 and 1, removing the time trend in net position sizethat reflects the growing VIX futures market over the sample period. The Appendix showsthat similar qualitative results hold using other proxies for risk and demand such as stockmarket returns and realized variance for risk and VIX ETP demand and the delta of VIXoptions traded by retail customers for demand.
The sample period is 2010 to 2018 using weekly observations on CoT release dates when
25
the DNP variable is reported. The sample period is motivated by several observations. First,the start date matches prior studies investigating the relationship between the pricing of VIXfutures and demand (Cheng 2018). During this period there are no breaks in the reportingof the DNP variable.5 In addition, the sample corresponds to a post-crisis period whenthe VIX ETP market is growing. Dong (2016) argues that the introduction of VIX ETPtrading represents a structural break in the VIX futures market, with ETPs introducingnew channels for demand to impact futures prices. Finally, the sample period is beneficialbecause a balanced panel is available to compute the average deviation measure across thefront six futures contracts.
Figure 8 presents the time-series relationship between the average deviation and therisk and demand variables that are used in the VAR. The top plot shows that the deviationmeasure decreases when the VIX increases. Since the deviation measure tracks the differencein prices for nearly identical claims across two markets, it is surprising that the no-arbitragedeviation responds to risk, as risk should play a similar role in both markets. One hypothesisfor what drives the result is that increases in risk may prevent traders from being able toengage in arbitrage trades that would drive prices back to fundamental values, perhaps as aresult of binding margin or value-at-risk constraints. Alternatively, hedgers may take profiton long positions when risk increases, leading to demand shocks that are correlated withincreases in risk and traders temporary inability to exploit arbitrage opportunities due tofinancing constraints. In addition to the correlation with risk, the bottom plot shows that thedeviation measure is also highly correlated with the DNP demand variable. This correlationmay be driven by systematic demand shocks as discussed above or by idiosyncratic shockssuch as mechanical roll effects from VIX ETPs. To more precisely identify how the no-arbitrage deviation measure relates to risk and demand and to better understand the lead-lag relationships, the paper estimates a VAR and studies its associated impulse responsefunctions.
6.2 VAR Impulse Response Functions
The paper estimates a trivariate VAR for the variables yt = [DEVt V IXt DNPt] at a weeklyfrequency from 2010 to 2018. Figure 9 reports impulse response functions (IRFs) from theVAR. The IRFs are from a Cholesky decomposition with the variables ordered as: VIX,DNP, DEV. The optimal lag length is selected by the SBIC criterion. The Appendix reportsthe IRFs with different orderings as a robustness check and finds qualitatively similar results.
5There is a gap in CoT report for VIX futures from December 2008 to June 2009 when open interest waslow and the position breakdown by trader type was not reported.
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The top row in Figure 9 reports the IRFs for DEV in response to VIX and DNP shocks.The top left plot shows that a one-standard deviation increase in VIX corresponds to a .25standard deviation decrease in DEV that mean reverts after four to six weeks. The topright plot shows that a one standard deviation increase in DNP corresponds to a .05 to.10 standard deviation increase in DEV that mean reverts over a longer horizon. Theseresponses are large in the sense that they represent several bid-ask spreads in VIX futuresand are the same order of magnitude as the coefficients on the deviation measure in thereturn predictability regressions. The impact of the VIX shock is about 3-4 times largerin magnitude than the impact of a demand shock over short horizons. Despite the tightrelationship between the deviation measure and demand in the time-series plot, the VARindicates that the risk shock is more significant over shorter horizons. For longer horizonsthe demand shock remains more significant and has a slightly larger magnitude than theVIX shock. Taken together, the results indicate that both the risk and demand channelshave an impact on the no-arbitrage deviation measure over different horizons.
The middle row in Figure 9 reveals how the DNP variable reacts to VIX and DEV shocks.The middle left plot shows that a one standard deviation increase in the VIX correspondsto a .10 decrease in DNP that persists for one to three months. The middle right plot showsthat a one standard deviation increase in DEV corresponds to an increase in DNP by .10standard deviations that peaks after one to two months and then persists over a longer periodof time. These results have a mixed interpretation. On one hand, DNP decreases when riskincreases. This result is consistent with dealers acting as hedgers that take profit on longpositions when risk increases. On the other hand, DNP increases when the deviation measureincreases. This result suggests that dealers also act as momentum traders, increasing theirlong position in VIX futures when the prices of VIX futures increase relative to variance swapforward rates. To the extent that the DNP variable is a veil for retail demand, the resultssuggest that retail traders use VIX ETPs to hedge volatility risk and chase momentum.
The bottom row Figure 9 reports how the VIX responds to DNP and DEV shocks. Theresponses are largely insignificant. This result provides a reassuring placebo test. One wouldnot expect changes in DNP or DEV to impact the VIX unless changes in these variablesled to arbitrage trading that moved index option prices and thus the VIX. For example,if DEV increases, arbitrage traders might sell VIX futures and pay fixed in variance swapforwards. Implementing the variance swap forward trade synthetically could put downwardpressure on the VIX if traders pay fixed in long-dated synthetic variance swaps by buyinglong-dated options and receive fixed in short-dated synthetic variance swap rates by sellingshort-dated options. The IRF in the bottom right plot provides some evidence that thereis a short-term negative effect of DEV shocks on the VIX over short horizons. Over longer
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horizons, however, the response becomes insignificant and the point estimate is close to zero.Given the large size of the index options market relative to the VIX futures market, theinsignificant responses that are consistent with the null hypothesis of no effect on the VIXseem most plausible.
6.3 Panel Regressions
Another way to study the deviation measure and its relationship with risk and demand is byexploiting a panel-regression approach. While the VAR accounts for the persistence of thedifferent variables and their joint interactions, it can be difficult to interpret large-scale VARs.This observation motivates estimating the VAR with only three variables, one of which is theaverage deviation measure across the front six contracts. In a panel-regression, the deviationmeasure for different contracts can be studied directly along with fixed effects to study withincontract variation and control variables to account for the economic environment.
Table 10 reports panel regressions that find similar relationships between the deviationmeasure and the risk and demand variables despite the different identification approach fromthe VAR. The regression specification is,
Deviationt+h,n = β∆xt+h + ρDeviationt,n+δControlst + FEs+εt+h,n. (11)
The horizon is one-week h = 5 to be similar to the VAR analysis. The regressions includeoverlapping observations from daily data using the front six contracts. The explanatoryvariable ∆xt+h and control variables are standardized but the deviation measure is not. Thesummary statistics for the deviation measure in Table 2 show that the deviation measure hasan average standard deviation of about .80%. This makes the results roughly comparable tothe IRFs from the VAR where the variables are standardized. The first three specifications 1-3 in Table 10 show how the deviation measure responds to changes in the explanatory variablecontrolling for the persistence of the deviation measure. The next three specifications 4-6repeat this analysis adding control variables and fixed effects. The control variables includethe time-to-maturity, initial margin, and open interest for the n-th contract on date t as wellas the lagged VIX index. Fixed effects are included for the contract, calendar year, and forthe contract being the first, second, etc. contract to maturity.
Panels A reports the results for the risk variables from 2007 to 2018. Panel B finds similarresults from 2010 to 2018, showing that the results are not driven by the financial crisis. Asin the VAR analysis, the results indicate that the deviation measure is decreasing in risk.When stock market returns are negative or volatility increases, the deviation measure tendsto decline. The magnitude of the point estimates is similar to the IRFs. For example, in the
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panel regressions, a one-standard deviation increase in the VIX is associated with a declinein the deviation measure of around -.15 in Panel A or -.18 in Panel B. In the VAR, the IRFof the deviation measure to a VIX shock has a point estimate of around -.10 to -.25 in thefirst few weeks after a shock. Despite the different approaches, the VAR and panel regressionreveal a similar relationship between the deviation measure and VIX index. Moreover, thepanel regressions highlight how the relationship between the deviation measure and risk isnot specific to the VIX index, but also holds for stock market returns and for the realizedvariance of the S&P 500 index.
Panel C reports the results for the demand variables from 2010 to 2018. The post-crisissample period is motivated by data availability. As in the VAR analysis, the deviationmeasure is increasing in demand pressure. A one-standard deviation increase in the dealernet position corresponds to a .05 standard-deviation increase in the deviation measure. Thisresponse is similar to the IRF in which the deviation measure increases by around .05 to .10in response to a DNP shock in the weeks following the shock. Beyond the DNP variable, theregressions show that the deviation measure is also increasing in VIX ETP net demand andin the delta of VIX options traded by retail customers. Similar to the DNP variable, theETP and option demand variables are normalized by open interest and then standardized.Compared to the risk variables in Panel B, the point estimates and explanatory power ofthe demand variables is slightly lower in Panel C. Over short horizons, changes in risk areassociated with larger changes in the deviation measure than changes in demand, similar tothe VAR analysis.
6.4 Other Channels
Beyond demand pressure, the limits-to-arbitrage literature highlights a number of additionalchannels that can lead to no-arbitrage violations and return predictability. For example,changes in margin requirements may lead to binding financial constraints, resulting in marginspirals if arbitrageurs are forced to unwind positions (Brunnermeier and Pedersen 2008).End-of-month and end-of-quarter dates can result in funding pressure that is associatedwith heightened arbitrage deviations in the foreign exchange market (Du et al. 2018). Salientevents like FOMC announcements may result in return predictability (Lucca and Moench2015). How does the no-arbitrage deviation measure studied in this paper relate to theseevents?
Figure 10 investigates this question by reporting event study plots to illustrate how thedeviation measure and VIX index respond to various events. The top row reports resultsfor margin changes. There are 41 increases and 33 decreases in the initial margin for the
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front month VIX futures contract from 2004 to 2018. In the two weeks leading up toa margin increase, the VIX index tends to increase by around four points. During thistime, the deviation measure decreases by around .20 units. Given the small number ofevents, this decline is not statistically significant. After the margin increase, the VIX tendsto decrease and the deviation measure tends to increase which is a decline in magnitude.From the plot, there does not appear to be a very strong response of the deviation measureto margin increases or decreases, aside from the typical reaction to changes in the VIX.The subsequent plots highlight similar null results for FOMC announcements, non-farmpayroll announcements, month-end dates, and quarter-end dates. There may be a smalldecrease in the deviation measure on the day following FOMC announcements, but otherwisethe deviation measure seems largely unrelated to these events. The Appendix providesrelated analysis for the risk and demand variables by highlighting how the deviation measureresponds to the largest changes in risk and demand over the sample period. While themagnitudes are different, the qualitative results are similar with changes in risk and demandbeing associated with the largest changes in the deviation measure.
7 Conclusion
This paper documents systematic law of one price deviations across the VIX futures andS&P 500 index options markets. The prices of VIX futures violate estimates of their no-arbitrage upper and lower bounds, indicating the presence of arbitrage opportunities. Ano-arbitrage deviation measure equal to the difference between the price of a VIX futurescontract and its corresponding synthetic variance swap forward rate is found to significantlypredicts returns. A relative value trading strategy that exploits the deviation measure earnsa significant Sharpe ratio with minimal exposure to traditional risk factors.
These results are surprising because the no-arbitrage relationships investigated in thepaper are well understood in the academic literature and should be known by option traders.Even if it is difficult to trade VIX futures against synthetic variance swap forwards in practice,there is still significant predictability when using the deviation measure to predict VIXfutures returns hedged with the stock market, and a sizeable Sharpe ratio and alpha-to-margin estimate persist even after taking VIX futures transaction costs into account. Thus,the results cannot be explained by a mere appeal to implementation challenges or transactioncosts.
Instead, the paper finds evidence that the no-arbitrage deviations are related to system-atic risk and demand pressure. Other channels like margin changes, economic announce-ments, and month-end effects are less significant in explaining the variation in the deviations
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over time. To the extent that dealer positions reflect retail demand, the results are consistentwith retail traders using VIX ETPs to hedge volatility risk and chase momentum, drivingthe prices of VIX futures away from fundamental values implied by the index option market.
An implication of these results is that investors and policymakers should be cautious wheninterpreting signals from equity volatility markets. Large no-arbitrage deviations indicatethat the VIX futures and index option markets are sending different messages about futurerisks.
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Table 1: Computing the Deviation Measure on Example Dates
This table illustrates two examples of how the paper computes the no-arbitrage deviation measureeach day. First, the paper obtains the variance swap curve in annualized variance units. Second, thepaper computes the variance swap forward curve assuming flat forward rates between the observedmaturities Fwdt,T1,T2 = (V St,T2 ·T2−V St,T1 ·T1)/(T2−T1). Third, the paper computes the differencebetween the VIX futures price and the one-month variance swap forward rate for the correspondingmaturity. The deviation measure Deviationt,n for the n-th contract is the difference between theVIX futures price Futt,n and the one-month variance swap forward rate Fwdt,n.
Example: Computation of Deviation Measure on 27Feb12Step 1: Variance swap rates (variance units)Maturity 1 2 3 6 9 12 15 18 24V St,T 3.39 3.90 4.55 5.83 6.44 6.88 7.18 7.52 8.16
Step 2: Variance swap forward rates (variance units)Maturity 0-1 1-2 2-3 3-6 6-9 9-12 12-15 15-18 18-24Fwdt,T1,T2
3.39 4.40 5.87 7.11 7.66 8.21 8.37 9.24 10.07
Step 3: Deviation measure Deviationt,n = Futt,n − Fwdt,n (volatility units)Maturity 21Mar 18Apr 16May 20Jun 18Jul 22Aug 19Sep 17Oct 21NovContract 1 2 3 4 5 6 7 8 9Futt,n 21.40 24.38 25.87 27.02 28.17 28.78 29.65 29.75 29.75Fwdt,n 20.35 23.23 25.67 26.67 26.67 27.48 27.67 27.67 28.47Deviationt,n 1.05 1.14 0.20 0.36 1.51 1.30 1.98 2.08 1.28
Example: Computation of Deviation Measure on 10Aug11Step 1: Variancee swap rates (variance units)Maturity 1 2 3 6 9 12 15 18 24V St,T 21.00 15.89 14.14 11.94 11.11 10.72 10.49 10.28 10.03
Step 2: Variance swap forward rates (variance units)Maturity 0-1 1-2 2-3 3-6 6-9 9-12 12-15 15-18 18-24Fwdt,T1,T2
21.00 10.77 10.65 9.73 9.45 9.56 9.58 9.24 9.28
Step 3: Deviation measure Deviationt,n = Futt,n − Fwdt,n (volatility units)Maturity 17Aug 21Sep 19Oct 16Nov 21Dec 18Jan 15Feb 21MarContract 1 2 3 4 5 6 7 8Futt,n 36.25 28.98 28.00 27.25 26.10 27.33 27.80 27.80Fwdt,n 43.23 32.75 32.23 31.20 31.20 31.07 30.74 30.74Deviationt,n -6.98 -3.78 -4.23 -3.95 -5.10 -3.74 -2.94 -2.94
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Table 2: Summary Statistics for Deviation Measure
This table reports summary statistics for the law of one price deviations between VIX futures andvariance swap forwards. The deviation measure is defined as Deviationt,n = Futt,n − Fwdt,n inannualized volatility units. Panel A reports summary statistics for the deviation variable. PanelB reports the correlation of the deviations across contracts. Panel C reports the average deviationby year for each contract. The sample period is 2007 to 2018 to obtain a balanced panel acrosscontracts.
Deviationt,n Summary StatisticsContract (n) (1) (2) (3) (4) (5) (6) Avg.
Panel A: Summary StatisticsMean -0.25 -0.47 -1.02 -1.26 -0.97 -1.08 -0.84Standard Deviation 0.82 0.74 0.77 0.78 0.88 0.93 0.82t-statistic -5.06 -7.34 -13.37 -15.38 -10.17 -11.07 -10.40Skewness -3.68 -1.72 0.32 -0.01 -0.83 -0.78 -1.12Kurtosis 31.49 14.07 5.50 3.13 5.11 5.80 10.85Minimum -8.87 -7.70 -4.89 -4.27 -5.86 -7.15 -6.4625th-Percentile -0.46 -0.78 -1.50 -1.75 -1.40 -1.60 -1.25Median -0.15 -0.41 -1.04 -1.26 -0.89 -1.03 -0.8075th-Percentile 0.13 -0.08 -0.52 -0.75 -0.40 -0.48 -0.35Maximum 4.26 4.42 4.22 2.16 1.58 1.44 3.01Autocorrelation 1-day 0.56 0.81 0.87 0.90 0.92 0.90 0.83Autocorrelation 5-day 0.33 0.65 0.73 0.80 0.84 0.78 0.69Autocorrelation 21-day 0.17 0.36 0.52 0.61 0.65 0.64 0.49
Panel B: Correlation MatrixDeviationt,1 1.00 0.38 0.06 0.06 0.20 0.19 0.18Deviationt,2 0.38 1.00 0.57 0.53 0.62 0.61 0.54Deviationt,3 0.06 0.57 1.00 0.72 0.53 0.55 0.49Deviationt,4 0.06 0.53 0.72 1.00 0.74 0.54 0.52Deviationt,5 0.20 0.62 0.53 0.74 1.00 0.83 0.58Deviationt,6 0.19 0.61 0.55 0.54 0.83 1.00 0.54
Panel C: Average Deviation by Year2007 -0.33 -0.26 -0.33 -0.39 -0.31 -0.31 -0.322008 -0.69 -0.57 -0.32 -0.66 -1.22 -1.41 -0.812009 -0.10 0.11 -0.47 -0.84 -1.12 -1.26 -0.612010 -0.44 -0.09 -0.48 -0.62 -0.25 -0.45 -0.392011 -0.38 -0.74 -1.38 -1.77 -1.33 -1.35 -1.162012 -0.15 -0.29 -1.04 -1.26 -0.23 -0.12 -0.522013 -0.09 -0.25 -0.82 -0.95 -0.39 -0.64 -0.532014 -0.12 -0.50 -1.26 -1.56 -1.12 -1.31 -0.982015 -0.27 -0.90 -1.66 -1.94 -1.64 -1.76 -1.362016 -0.14 -0.66 -1.36 -1.58 -1.18 -1.36 -1.052017 -0.02 -0.69 -1.55 -1.84 -1.22 -1.32 -1.112018 -0.21 -0.85 -1.55 -1.73 -1.57 -1.62 -1.25
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Table 3: The Frequency of the Law of One Price Violations
This table reports the frequency of law of one price violations for VIX futures contracts from theirno-arbitrage bounds. Panel A reports the frequency of upper bound violations over the full samplefrom 2004 to 2018 (A.I) and during a post-crisis sample 2010 to 2018 (A.II). An upper boundviolation occurs when the deviation measure is greater than zero, Deviationt,n > 0. The table alsoreports the frequency of violations that are greater than thresholds of .25% and .50%. Panel Breports the analogous frequency of lower bound violations. The lower bound is estimated for eachcontract-day as the upper bound minus the difference between the upper bound and lower boundfrom a term-structure model for that contract-day pair. The results indicate that around 20% ofcontract-date observations exhibit an upper bound or lower bound violations on average.
Frequency of Law of One Price ViolationsContract (n) (1) (2) (3) (4) (5) (6) Avg.
Panel A.I: VIX Futures - Upper Bound > Threshold 2004-2018Threshold = 0 0.35 0.24 0.13 0.14 0.14 0.14 0.19Threshold = .25 0.17 0.13 0.07 0.08 0.09 0.09 0.11Threshold = .50 0.07 0.06 0.03 0.05 0.06 0.06 0.05
Panel A.II: VIX Futures - Upper Bound > Threshold 2010-2018Threshold = 0 0.36 0.15 0.03 0.02 0.09 0.09 0.12Threshold = .25 0.15 0.07 0.02 0.01 0.05 0.06 0.06Threshold = .50 0.06 0.03 0.00 0.00 0.03 0.04 0.03
Panel B.I: VIX Futures - Lower Bound < Threshold 2004-2018Threshold = 0 0.11 0.12 0.31 0.35 0.19 0.21 0.22Threshold = -.25 0.06 0.06 0.21 0.25 0.12 0.13 0.14Threshold = -.50 0.03 0.03 0.12 0.15 0.08 0.08 0.08
Panel B.II: VIX Futures - Lower Bound < Threshold 2010-2018Threshold = 0 0.08 0.16 0.49 0.55 0.25 0.28 0.30Threshold = -.25 0.04 0.08 0.33 0.40 0.15 0.17 0.20Threshold = -.50 0.03 0.04 0.19 0.25 0.10 0.10 0.12
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Table 4: Summary Statistics for VIX Futures and Variance Swap Forwards
Panel A reports summary statistics for VIX futures prices for the front six contracts and for varianceswap one-month forward rates for the corresponding maturities. Panel B reports summary statisticsfor VIX futures and variance swap forward one-week excess returns (h = 5). The sample period is2007 to 2018 for a balanced panel.
VIX Futures and VS Forwards Summary StatisticsContract (n) (1) (2) (3) (4) (5) (6) AveragePanel A.I: VIX Futures Prices Futt,nMean 20.07 20.81 21.30 21.65 21.94 22.20 21.33Standard Deviation 8.57 7.66 7.06 6.60 6.28 6.02 7.03Skewness 2.17 1.82 1.59 1.38 1.21 1.10 1.54Kurtosis 9.04 7.21 6.20 5.10 4.33 3.90 5.96Median 17.54 18.58 19.27 19.67 20.08 20.27 19.23
Panel A.II: VS Forward Rates Fwdt,n (annualized volatility units)Mean 20.32 21.28 22.32 22.91 22.91 23.28 22.17Standard Deviation 8.80 7.74 6.91 6.48 6.47 6.21 7.10Skewness 2.21 1.90 1.62 1.41 1.41 1.32 1.65Kurtosis 9.50 7.66 6.28 5.21 5.18 4.86 6.45Median 17.82 19.28 20.64 21.27 21.30 21.62 20.32
Panel B.I: VIX Futures Returns (percent) RV IXt+h,n = Futt,n − Futt+h,nMean 0.17 0.15 0.09 0.06 0.06 0.05 0.10Standard Deviation 2.42 1.99 1.57 1.34 1.19 1.10 1.60Sharpe Ratio 0.07 0.08 0.06 0.05 0.05 0.05 0.06t-statistic 1.81 1.89 1.48 1.19 1.24 1.23 1.47Skewness -1.56 -1.20 -1.03 -0.86 -0.80 -0.74 -1.03Kurtosis 12.77 10.53 9.42 8.21 8.29 7.35 9.43Median 0.27 0.30 0.20 0.15 0.10 0.10 0.19
Panel B.I: VS Forward Returns (basis points) RV SFt+h,n = Fwdt,n − Fwdt+h,nMean 0.05 0.50 1.01 0.80 -0.38 -0.09 0.31Standard Deviation 18.98 12.37 9.54 8.21 6.75 6.10 10.33Sharpe Ratio 0.00 0.04 0.11 0.10 -0.06 -0.02 0.03t-statistic 0.07 0.99 2.44 2.23 -1.54 -0.45 0.62Skewness -3.84 -2.82 -2.84 -2.81 -0.30 -0.09 -2.12Kurtosis 67.11 45.54 30.56 28.92 15.93 12.59 33.44Median 1.17 1.07 1.63 1.18 -0.32 0.11 0.81
38
Table 5: Deviation Measure Predicts the Returns of VIX FuturesHedged with Variance Swap Forwards
This table reports return predictability regressions for hedged VIX futures returns over a weeklyhorizon (h=5). The columns report results for each of the front six contracts. The first step regressesVIX futures returns onto variance swap forward returns to estimate the hedge ratio β̂n. The secondstep regresses the hedged return onto the deviation measure. The deviation measure significantlypredicts hedged returns across contracts and sample periods, and is robust to the presence of otherpredictors like the VIX and realized variance (RV) which proxy for the variance risk premium. Thevariables in the second step regression are z-scored for ease of interpretation. Panel A (B) reportsresults for the full (post-crisis) sample.
Return Predictability Regression: RV IXt+h,n − β̂nRV SFt+h,n = γ′nxt,n + εt+h,n
Panel A.I: Full sample from 2004 to 2018Contract (n) (1) (2) (3) (4) (5) (6)Deviation 0.19∗∗∗ 0.23∗∗∗ 0.27∗∗∗ 0.25∗∗∗ 0.36∗∗∗ 0.38∗∗∗
(0.03) (0.05) (0.05) (0.03) (0.07) (0.08)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.038 0.051 0.075 0.063 0.129 0.145
Panel A.II: Full sample from 2004 to 2018 with controlsContract (n) (1) (2) (3) (4) (5) (6)Deviation 0.27∗∗∗ 0.31∗∗∗ 0.33∗∗∗ 0.32∗∗∗ 0.37∗∗∗ 0.39∗∗∗
(0.04) (0.04) (0.05) (0.04) (0.05) (0.07)VIX 0.29∗∗∗ -0.00 0.01 -0.03 0.07 0.23∗∗
(0.11) (0.15) (0.12) (0.13) (0.11) (0.11)RV -0.13 0.09 0.14 0.20 -0.16 -0.23∗∗
(0.11) (0.19) (0.16) (0.16) (0.12) (0.09)RMRF -0.01 -0.09∗∗ -0.02 -0.11∗∗∗ -0.02 0.03
(0.04) (0.04) (0.05) (0.04) (0.05) (0.05)VLM -0.01 0.13∗∗∗ 0.16∗∗∗ 0.11∗∗∗ 0.17∗∗∗ 0.16∗∗∗
(0.03) (0.03) (0.03) (0.04) (0.04) (0.03)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.065 0.076 0.117 0.111 0.175 0.175
Panel B.I: Post-crisis sample from 2010 to 2018Contract (n) (1) (2) (3) (4) (5) (6)Deviation 0.23∗∗∗ 0.34∗∗∗ 0.28∗∗∗ 0.26∗∗∗ 0.40∗∗∗ 0.38∗∗∗
(0.04) (0.03) (0.04) (0.04) (0.05) (0.05)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.052 0.112 0.078 0.068 0.158 0.143
Panel B.II: Post-crisis sample from 2010 to 2018 with controlsContract (n) (1) (2) (3) (4) (5) (6)Deviation 0.29∗∗∗ 0.39∗∗∗ 0.32∗∗∗ 0.30∗∗∗ 0.40∗∗∗ 0.37∗∗∗
(0.04) (0.05) (0.04) (0.05) (0.05) (0.05)VIX 0.26∗∗∗ 0.11 -0.00 -0.12 -0.00 0.07
(0.09) (0.10) (0.11) (0.10) (0.10) (0.10)RV -0.09 -0.04 0.06 0.10 -0.03 -0.04
(0.08) (0.08) (0.09) (0.08) (0.09) (0.09)RMRF 0.04 0.00 -0.02 -0.03 0.02 0.07
(0.05) (0.04) (0.05) (0.05) (0.05) (0.05)VLM -0.02 0.15∗∗ 0.20∗∗∗ 0.09∗∗ 0.05 0.08∗∗
(0.04) (0.06) (0.04) (0.04) (0.04) (0.04)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.079 0.136 0.118 0.077 0.160 0.150
Newey-West SEs with 15 lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
39
Table 6: Deviation Measure Predicts the Returns of VIX FuturesHedged with Stock Market Returns
This table reports return predictability regressions for VIX futures returns that are hedged withstock market returns over a weekly horizon (h=5). The regressions are analogous to Table 5. Whenhedging with stock market returns, the deviation measure remains significant at predicting VIXfutures returns, particularly during the post-crisis period. These results provide evidence that thedeviation measure is identifying mispricing of VIX futures relative to variance swap forwards.
Return Predictability Regression: RV IXt+h,n − βnRMRFt+h,n = γ′nxt,n + εt+h,n
Panel A.I: Full sample from 2004 to 2018Contract (n) (1) (2) (3) (4) (5) (6)Deviation 0.24∗∗∗ 0.28∗∗∗ 0.04 0.07∗∗ 0.17∗∗∗ 0.16∗∗∗
(0.07) (0.07) (0.04) (0.03) (0.06) (0.06)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.055 0.079 0.001 0.005 0.029 0.026
Panel A.II: Full sample from 2004 to 2018 with controlsContract (n) (1) (2) (3) (4) (5) (6)Deviation 0.28∗∗∗ 0.28∗∗∗ 0.05 0.06∗ 0.13∗∗∗ 0.12∗∗∗
(0.06) (0.06) (0.04) (0.04) (0.04) (0.04)VIX 0.43∗∗∗ 0.23 0.18 0.13 0.06 0.07
(0.12) (0.17) (0.17) (0.16) (0.16) (0.16)RV -0.36∗∗∗ -0.30 -0.36∗ -0.30 -0.17 -0.18
(0.12) (0.19) (0.20) (0.19) (0.17) (0.16)RMRF 0.03 -0.01 -0.00 0.01 -0.00 -0.02
(0.07) (0.07) (0.07) (0.07) (0.08) (0.07)VLM -0.02 0.09∗∗∗ 0.02 0.01 0.04 0.06∗
(0.04) (0.03) (0.03) (0.04) (0.04) (0.03)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.075 0.104 0.041 0.037 0.043 0.044
Panel B.I: Post-crisis sample from 2010 to 2018Contract (n) (1) (2) (3) (4) (5) (6)Deviation 0.14∗∗ 0.23∗∗∗ 0.16∗∗∗ 0.16∗∗∗ 0.18∗∗∗ 0.15∗∗∗
(0.06) (0.06) (0.05) (0.05) (0.06) (0.06)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.019 0.053 0.025 0.025 0.031 0.022
Panel B.II: Post-crisis sample from 2010 to 2018 with controlsContract (n) (1) (2) (3) (4) (5) (6)Deviation 0.22∗∗∗ 0.27∗∗∗ 0.14∗∗∗ 0.13∗∗∗ 0.15∗∗∗ 0.12∗∗
(0.05) (0.06) (0.04) (0.05) (0.05) (0.05)VIX 0.30∗∗∗ 0.15 0.06 0.02 -0.01 -0.01
(0.10) (0.12) (0.12) (0.12) (0.11) (0.11)RV -0.12 -0.08 -0.11 -0.09 -0.05 -0.06
(0.10) (0.11) (0.12) (0.12) (0.11) (0.11)RMRF 0.01 -0.02 0.00 0.01 0.01 0.03
(0.06) (0.05) (0.05) (0.05) (0.05) (0.06)VLM -0.03 0.05 -0.05 -0.05 -0.03 0.01
(0.05) (0.05) (0.04) (0.05) (0.05) (0.04)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.054 0.063 0.029 0.032 0.033 0.027
Newey-West SEs with 15 lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
40
Table 7: Deviation is Less Significant for Variance Swap Forward ReturnsHedged with the Stock Market, Especially in Post-Crisis Period
This table reports return predictability regressions for variance swap forwards that are hedged withstock market returns over a weekly horizon (h=5). The regressions are analogous to Table 5. Incomparison to VIX futures, the deviation measure is less significant at predicting hedged varianceswap forward returns, particularly during the post-crisis period where there is lack of significancefor some contracts. These results provide less evidence that the deviation measure is identifyingmispricing of variance swap forwards relative to VIX futures. That said, the deviation measure doespredict hedged variance swap forward returns for most contracts and it tends to have the expected,negative sign, indicating that the evidence is somewhat mixed.
Return Predictability Regression: RV SFt+h,n − βnRMRFt+h,n = γ′nxt,n + εt+h,n
Panel A.I: Full sample from 2004 to 2018Contract (n) (1) (2) (3) (4) (5) (6)Deviation 0.05 0.13 -0.15∗∗∗ -0.11∗∗∗ -0.11∗∗ -0.16∗∗∗
(0.09) (0.09) (0.05) (0.03) (0.06) (0.05)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.002 0.016 0.022 0.012 0.012 0.025
Panel A.II: Full sample from 2004 to 2018 with controlsContract (n) (1) (2) (3) (4) (5) (6)Deviation 0.03 0.07 -0.17∗∗∗ -0.16∗∗∗ -0.15∗∗∗ -0.21∗∗∗
(0.08) (0.06) (0.05) (0.04) (0.05) (0.06)VIX 0.34∗∗ 0.33 0.29 0.26 0.11 -0.01
(0.15) (0.23) (0.20) (0.20) (0.17) (0.15)RV -0.45∗∗ -0.46 -0.57∗∗ -0.55∗∗ -0.15 -0.10
(0.23) (0.30) (0.28) (0.27) (0.23) (0.17)RMRF 0.04 0.04 0.00 0.09 0.01 -0.06
(0.06) (0.08) (0.06) (0.08) (0.10) (0.08)VLM 0.02 0.02 -0.08∗∗∗ -0.05∗∗ -0.09∗∗∗ -0.06∗
(0.03) (0.03) (0.03) (0.02) (0.03) (0.03)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.032 0.048 0.123 0.115 0.019 0.038
Panel B.I: Post-crisis sample from 2010 to 2018Contract (n) (1) (2) (3) (4) (5) (6)Deviation -0.12∗∗∗ 0.00 -0.04 -0.02 -0.12∗∗ -0.16∗∗∗
(0.04) (0.06) (0.05) (0.06) (0.05) (0.05)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.014 -0.000 0.001 0.000 0.015 0.026
Panel B.II: Post-crisis sample from 2010 to 2018 with controlsContract (n) (1) (2) (3) (4) (5) (6)Deviation -0.05 0.01 -0.08∗ -0.07 -0.14∗∗∗ -0.18∗∗∗
(0.05) (0.06) (0.04) (0.06) (0.05) (0.05)VIX 0.16∗ 0.11 0.11 0.14 0.03 -0.03
(0.10) (0.12) (0.13) (0.13) (0.12) (0.11)RV -0.09 -0.08 -0.18 -0.20∗ -0.05 -0.07
(0.10) (0.11) (0.12) (0.12) (0.10) (0.10)RMRF -0.06 -0.03 0.01 0.03 -0.02 -0.04
(0.05) (0.06) (0.06) (0.06) (0.06) (0.06)VLM -0.00 -0.04 -0.18∗∗∗ -0.11∗∗ -0.06 -0.04
(0.04) (0.05) (0.04) (0.05) (0.04) (0.04)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.025 0.005 0.037 0.016 0.017 0.036
Newey-West SEs with 15 lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
41
Table 8: Summary Statistics for Relative Value Trading Strategies
This table reports summary statistics for weekly returns from the relative value trading strategiesbased on the deviation measure in comparison to stock market and volatility factor returns. PanelA reports statistics for the relative value strategy using VIX futures returns hedged with varianceswap forward returns (1), stock market returns (2), and stock market returns including transactioncosts (3). Columns (4-6) report results for a post-crisis sample from 2010 to 2018. Panel B reportsanalogous statistics for stock market returns (RMRF), receiving fixed in one-month variance swapforwards (VS1), and selling the front month VIX futures contract (VX1).
Weekly Return Summary Statistics
Panel A: VIX futures trading strategy levered for 10% annualized volatilitySpecification (1) (2) (3) (4) (5) (6)Hedge VSF RMRF RMRF VSF RMRF RMRFTransaction Costs No No Yes No No YesPost-Crisis No No No Yes Yes YesMean 0.58 0.27 0.14 0.54 0.21 0.16Standard Deviation 1.39 1.39 1.39 1.39 1.39 1.39Sharpe Ratio 0.42 0.20 0.10 0.39 0.15 0.12t-statistic 14.54 6.29 3.26 10.69 3.81 2.98Skewness 0.71 1.41 2.09 1.25 1.61 2.03Kurtosis 18.44 13.07 20.62 26.12 17.32 23.82Minimum -11.58 -7.04 -8.50 -9.64 -7.27 -9.6725th-Percentile -0.06 -0.39 -0.27 -0.08 -0.40 -0.23Median 0.51 0.16 0.00 0.51 0.13 0.0075th-Percentile 1.15 0.84 0.50 1.12 0.75 0.54Maximum 16.48 13.91 15.17 18.02 14.36 17.27Negative Percent 26.66 39.99 32.64 27.29 41.94 31.30Sortino Ratio 0.87 0.42 0.20 0.79 0.30 0.23Maximum Drawdown 6.72 9.67 16.68 6.30 9.96 8.17Leverage (percentage) 11.62 7.87 5.78 12.71 8.12 6.57CAPM α-to-margin 4.90 3.61 2.64 4.12 2.74 2.75Observations 3698 3698 3698 2246 2246 2246
Panel B: Stock market, variance swap, and VIX futures factor returnsSpecification (1) (2) (3) (4) (5) (6)Return RMRF VS1 VX1 RMRF VS1 VX1Post-Crisis No No No Yes Yes YesMean 0.17 0.03 0.19 0.24 0.03 0.23Standard Deviation 2.37 0.21 2.20 2.08 0.14 2.14Sharpe Ratio 0.07 0.15 0.09 0.11 0.20 0.11t-statistic 2.21 4.19 2.54 2.90 5.40 2.68Skewness -0.92 -0.97 -1.71 -0.80 -1.98 -1.29Kurtosis 11.74 68.08 15.17 6.57 30.99 12.50Minimum -19.81 -3.03 -21.04 -14.47 -1.43 -19.7025th-Percentile -0.86 -0.00 -0.40 -0.71 -0.00 -0.45Median 0.32 0.03 0.21 0.38 0.03 0.2575th-Percentile 1.44 0.07 1.15 1.45 0.06 1.25Maximum 18.26 2.84 13.02 9.00 0.98 10.35Negative Percent 41.24 26.04 33.59 39.31 26.40 33.53Sortino Ratio 0.11 0.21 0.12 0.18 0.29 0.16Maximum Drawdown 50.76 5.74 45.03 16.38 1.18 16.69Observations 3698 3698 3698 2246 2246 2246
42
Table 9: Alpha-to-Margin Estimates for the Relative Value Trading Strategies
This table reports weekly factor return regressions for the relative value trading strategies using themaximum leverage to required initial margin for VIX futures. The alpha-to-margin estimates arelarge and significant across hedging portfolios, factor model specifications, and sample periods.
Factor Model CAPM FFC4 FFCV6 CAPM FFC4 FFCV6Specification (1) (2) (3) (4) (5) (6)Panel A: Variance Swap HedgeAlpha 4.90∗∗∗ 4.92∗∗∗ 4.83∗∗∗ 4.12∗∗∗ 4.16∗∗∗ 4.02∗∗∗
(0.35) (0.35) (0.39) (0.43) (0.44) (0.59)RMRF 0.37 0.28 0.01 0.44 0.32 -0.09
(0.32) (0.38) (0.35) (0.42) (0.47) (0.43)HML -0.06 -0.04 0.02 -0.04
(0.26) (0.25) (0.39) (0.38)SMB 0.32 0.36 0.55 0.60
(0.36) (0.35) (0.47) (0.47)MOM -0.14 -0.11 -0.16 -0.22
(0.19) (0.19) (0.30) (0.29)VS1 4.06 9.42
(4.52) (16.40)VX1 0.04 -0.07
(0.44) (0.82)Adjusted R2 0.005 0.006 0.008 0.006 0.009 0.013Panel B: Stock Market HedgeAlpha 3.61∗∗∗ 3.65∗∗∗ 4.00∗∗∗ 2.74∗∗∗ 2.85∗∗∗ 3.23∗∗∗
(0.54) (0.55) (0.55) (0.72) (0.72) (0.72)RMRF -0.74∗∗ -0.92∗∗ 1.21∗∗ -0.78 -0.89 2.12∗∗∗
(0.31) (0.39) (0.50) (0.56) (0.62) (0.61)HML -0.45 -0.50 -1.60∗∗ -1.28∗∗∗
(0.46) (0.40) (0.63) (0.49)SMB 0.60 0.31 0.91 0.64
(0.56) (0.51) (0.78) (0.66)MOM -0.55∗ -0.54∗ -1.59∗∗∗ -1.38∗∗∗
(0.33) (0.30) (0.54) (0.46)VS1 -7.93∗ -14.90
(4.58) (9.92)VX1 -2.38∗∗∗ -2.96∗∗∗
(0.71) (0.93)Adjusted R2 0.010 0.014 0.069 0.009 0.034 0.119Panel C: Stock Market Hedge with Transaction CostsAlpha 2.64∗∗∗ 2.72∗∗∗ 3.24∗∗∗ 2.75∗∗∗ 2.93∗∗∗ 3.55∗∗∗
(0.74) (0.75) (0.73) (0.87) (0.87) (0.81)RMRF -0.98∗∗ -1.40∗∗∗ 1.98∗∗∗ -1.20∗ -1.38∗ 2.97∗∗∗
(0.39) (0.52) (0.73) (0.68) (0.76) (0.81)HML -0.28 -0.35 -1.93∗∗ -1.45∗∗
(0.70) (0.59) (0.79) (0.60)SMB 1.48∗∗ 1.02 1.35 0.96
(0.74) (0.64) (0.96) (0.80)MOM -0.79∗ -0.76∗∗ -2.24∗∗∗ -1.92∗∗∗
(0.42) (0.37) (0.62) (0.50)VS1 -10.48 -27.37∗∗
(7.88) (11.13)VX1 -3.95∗∗∗ -3.91∗∗∗
(1.13) (1.40)Adjusted R2 0.009 0.016 0.093 0.013 0.045 0.155Post-Crisis No No No Yes Yes YesObservations 3698 3698 3698 2246 2246 2246Newey-West SEs with 15 lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
43
Table 10: The No-Arbitrage Deviation Measure is Decreasing in Riskand Increasing in Demand Pressure for VIX Futures
This table reports a panel regression of the deviation measure onto changes in different risk anddemand variables over a one-week horizon h = 5. Specifications 1-3 control for the lagged deviation.Specifications 4-6 add controls variables and fixed effects. The control variables include the contract-specific time-to-maturity, open interest, and initial margin and the lagged VIX index to proxy forthe economic environment. The fixed effects include the contract, contract number, and calendaryear. The explanatory variables are z-scored for ease of interpretation. Across the different variablesproxying for risk and demand and regression specifications, the results indicate that the deviationmeasure is decreasing in risk and increasing in demand.
Deviationt+h,n = β∆xt+h + ρDeviationt,n + δControlst + FEs + εt+h,n
Panel A: Risk Factors from 2007 to 2018Specification (1) (2) (3) (4) (5) (6)Explanatory Variable RMRF RV VIX RMRF RV VIX∆xt+h 0.10∗∗∗ -0.05∗∗∗ -0.14∗∗∗ 0.11∗∗∗ -0.25∗∗∗ -0.15∗∗∗
(0.02) (0.02) (0.02) (0.02) (0.05) (0.02)Deviationt,n 0.79∗∗∗ 0.79∗∗∗ 0.81∗∗∗ 0.60∗∗∗ 0.61∗∗∗ 0.60∗∗∗
(0.02) (0.02) (0.02) (0.03) (0.03) (0.03)Observations 17689 17689 17689 17689 17689 17689Adjusted R2 0.655 0.646 0.667 0.694 0.695 0.707Controls and FEs No No No Yes Yes Yes
Panel B: Risk Factors from 2010 to 2018Specification (1) (2) (3) (4) (5) (6)Explanatory Variable RMRF RV VIX RMRF RV VIX∆xt+h 0.17∗∗∗ -0.06∗∗ -0.17∗∗∗ 0.17∗∗∗ -0.33∗∗∗ -0.18∗∗∗
(0.02) (0.02) (0.02) (0.02) (0.05) (0.02)Deviationt,n 0.85∗∗∗ 0.84∗∗∗ 0.86∗∗∗ 0.66∗∗∗ 0.64∗∗∗ 0.65∗∗∗
(0.02) (0.02) (0.01) (0.03) (0.03) (0.03)Observations 13260 13260 13260 13260 13260 13260Adjusted R2 0.749 0.724 0.756 0.789 0.782 0.794Controls and FEs No No No Yes Yes Yes
Panel C: Demand Factors from 2010 to 2018Specification (1) (2) (3) (4) (5) (6)Explanatory Variable Dealer ETP Option Dealer ETP Option∆xt+h 0.04∗∗∗ 0.02 0.05∗∗∗ 0.05∗∗∗ 0.03∗∗ 0.07∗∗∗
(0.01) (0.01) (0.01) (0.01) (0.01) (0.01)Deviationt,n 0.85∗∗∗ 0.85∗∗∗ 0.85∗∗∗ 0.67∗∗∗ 0.67∗∗∗ 0.67∗∗∗
(0.02) (0.02) (0.02) (0.03) (0.03) (0.03)Observations 13260 13260 13260 13260 13260 13260Adjusted R2 0.724 0.723 0.725 0.765 0.764 0.767Controls and FEs No No No Yes Yes Yes
SEs double-clustered by date and contract, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
44
Figure 1: Equity Volatility Market Size and Dealer Position
96 98 00 02 04 06 08 10 12 14 16 18
Date
0
1000
2000
3000
4000
5000
$ M
illio
n
SPX Index Options vs. VIX Futures Market Size
SPX Option Open Interest Black-Scholes VegaVIX Futures Open Interest Dollar per Point
04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
Date
0
100
200
300
400
500
600
700
800
$ M
illio
n
VIX Futures vs. VIX ETP Market Size
VIX Futures Open Interest Dollar per PointVIX Short-Term ETP Total Vega
09 10 11 12 13 14 15 16 17 18 19
Date
-200
-100
0
100
200
300
$ M
illio
n
VIX Futures Demand Variables: Correlation = 61%
CoT Dealer Net Position in VIX FuturesVIX Short-Term ETP Net Vega
06 07 08 09 10 11 12 13 14 15 16 17 18 19
Date
-80
-60
-40
-20
0
20
40
60
80
Perc
ent of O
pen Inte
rest
CoT Net Position for Different Trader Types
Dealer
Levered Fund: Corr(D,L) =-91%
Asset Manager: Corr(D,A) =-4%
Other Reportable: Corr(D,O) =-53%
This figure provides an overview of equity volatility market size and investor positioning. TheS&P 500 index options and VIX futures markets have experienced substantial growth over the lastdecade. The index options market has been much larger than the VIX futures market throughoutthe sample (top left). The growth in the VIX futures market has coincided with growth in theVIX ETP market starting in 2009 (top right). Focusing on the behavior of certain traders, dealerpositions in VIX futures are highly related to demand for VIX ETPs (bottom left). Within the VIXfutures market, dealers and leveraged funds generally take large and opposing positions (bottomright). The break in the time-series of CoT positions corresponds to a period in early 2009 whenVIX futures open interest was low and the breakdown was not reported. One interpretation of theposition data is that retail demand for ETPs is hedged by dealers against leveraged funds.
45
Figure 2: Variance Swap Forward Rate Estimation
0 3 6 9 12 15 18 21 24
Time-to-Maturity (Months)
3.00
4.00
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7.00
8.00
9.00
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11.00
Price
(A
nn
ua
lize
d V
aria
nce
Un
its)
Variance Swap Forward Curve on 02/27/12
VS Rate VSt,T
VS Instantaneous Forward Rate Ft,T,T
VS 1mn Forward Rate Fwdt,T,T+1
0 3 6 9 12 15 18 21 24
Time-to-Maturity (Months)
8.00
10.00
12.00
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22.00
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(A
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Un
its)
Variance Swap Forward Curve on 08/10/11
VS Rate VSt,T
VS Instantaneous Forward Rate Ft,T,T
VS 1mn Forward Rate Fwdt,T,T+1
This figure illustrates the computation of one-month variance swap forward rates for the deviationmeasure. The variance swap curve in blue is decomposed into an instantaneous forward curve ingreen assuming flat forward rates between the observed maturities. The one-month forward rates atVIX futures maturity dates are an average of the green variance swap forward curve over the nextmonth. Variance swap rates and forward rates in these plots are expressed in annualized varianceunits.
46
Figure 3: Examples of Law of One Price Violations
0 1 2 3 4 5 6 7 8 9 10
Contract Number
20
22
24
26
28
30
Price
Futures Prices vs. No-Arbitrage Bounds on 02/27/12
VIX Futures Price: Futt,n
Variance Swap Forward Rate: Fwdt,n
(Upper Bound)
0 1 2 3 4 5 6 7 8 9 10
Contract Number
26
28
30
32
34
36
38
40
42
44
Price
Futures Prices vs. No-Arbitrage Bounds on 08/10/11
VIX Futures Price: Futt,n
Variance Swap Forward Rate: Fwdt,n
(Upper Bound)
Volatility Swap Rate: Fwdt,n
- (UBts
t,n - LB
ts
t,n) (Lower Bound)
The deviation measure is the difference between the VIX futures price in blue and one-monthvariance swap forward rate in red both expressed in annualized volatility units. The top plotillustrates examples of static arbitrage opportunities in which the prices of VIX futures are abovethe upper bound. An arbitrageur could lock in a riskless profit with no capital outlay by selling theexpensive VIX futures contracts with the hedge ratio from the paper, paying fixed in the varianceswap forwards, and holding the position until maturity. The bottom plot illustrates an examplewhen the prices of VIX futures are significantly below the upper bound and an estimate of thelower bound. In this case, an arbitrageur could lock in a profit by buying VIX futures and receivingfixed in volatility swap forwards. The lower bound is the upper bound in red minus the differencebetween an estimate of the upper bound and lower bound from a term-structure model.
47
Figure 4: Deviation of VIX Futures from the Law of One Price
This figure plots the average deviation and average absolute deviation for the front six contractsfrom March 2004 to December 2018. The deviation measure for the n-th contract is the differencebetween the VIX futures price and its no-arbitrage upper bound, Deviationt,n = Futt,n − Fwdt,n.Positive values are law of one price violations. Gray shading indicates NBER recessions.
48
Figure 5: Deviation of the Front Contract from the Law of One Price
This figure plots the deviation measure for the front contract alongside the VIX futures and varianceswap forward prices that are used to compute the deviation measure. While the futures price andforward price tend to track each other closely, there are periods with prolonged and significant lawof one price deviations during the sample period.
49
Figure 6: Performance of Relative Value Strategy Based on Deviation Measure
04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
Date
0
5
10
15
20
25
Cum
ula
tive S
um
of W
eekly
Retu
rns
NBER Recession
Relative Value Returns: Sharpe Ratio = 3.00
Stock Market Returns: Sharpe Ratio = 0.51
This figure plots the performance of the relative value trading strategy in VIX futures and varianceswap forwards against stock market returns. The relative value strategy goes long (short) VIXfutures when the deviation measure exceeds a low (high) threshold of τ = .50 z-scores for the frontsix contracts and hedges with variance swap forwards. Each trade is held for a one-week horizon,with the strategy forming an equally-weighted portfolio when multiple contracts are traded on thesame day. The plot reports the cumulative sum of weekly returns for the strategy and stock marketwhich are normalized to 10% annualized volatility for comparison. The relative value strategy earnsa large Sharpe ratio and exhibits low drawdowns compared to the stock market.
50
Figure 7: Robustness of Relative Value Trading Strategy
04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
Date
0
5
10
15
20
25
Cum
ula
tive S
um
of W
eekly
Retu
rns
Performance of Relative Value Trading Strategies
NBER Recession
Strategy with VS Fwd Hedge: Sharpe Ratio = 3.00
Strategy with Stock Market Hedge: Sharpe Ratio = 1.43
Strategy with Stock Market Hedge less T-Costs: Sharpe Ratio = 0.73
Stock Market Returns: Sharpe Ratio = 0.51
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0
0.2
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0.8
1
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Sharp
e R
atio (
Annualiz
ed)
Contract 1: 2004-2018
VS Fwd Hedge
Stock Market Hedge
Stock Market Hedge with T-Costs
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0
0.5
1
1.5
2
Sharp
e R
atio (
Annualiz
ed)
Contracts 1-2: 2004-2018
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0
0.5
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Sharp
e R
atio (
Annualiz
ed)
Contracts 1-4: 2004-2018
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0
0.5
1
1.5
2
2.5
3
3.5
Sharp
e R
atio (
Annualiz
ed)
Contracts 1-6: 2004-2018
This figure illustrates the robustness of the relative value trading strategy to hedging with stockmarket returns, including transaction costs, trading different numbers of contracts, and varying thedeviation-based trading threshold. The top plot reports the time-series of cumulative returns for thebaseline strategies discussed in the paper in comparison to the stock market. The plot normalizesthe annualized volatility of each series to 10% for comparison. The bottom plot reports the Sharperatios for the baseline strategies varying the number of contracts traded and threshold for trading.Across specifications, the relative value trading strategy earns a large Sharpe ratio and producesreturns that are largely uncorrelated with traditional risk factors.
51
Figure 8: Time-Series of Deviation, VIX, and Dealer Net Position for VAR
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5Z
-Score
Deviation vs. VIX Index: Correlation = -24%
Average Deviation
VIX Index
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-S
core
Deviation vs. CoT Dealer Net Position: Correlation = 57%
Average Deviation
CoT Dealer Net Position
This figure plots the average deviation measure against the VIX index and dealer net position at aweekly frequency from 2010 to 2018 (T = 469). The VAR is estimated using these three variables,yt =[Deviationt VIXt Dealer Net Positiont], which are standardized for comparison and ease ofinterpretation. The deviation measure is negatively correlated with the VIX index which serves as aproxy for risk (top plot). The deviation measure is positively correlated with the dealer net positionwhich serves as a proxy for demand to buy VIX futures (bottom plot).
52
Figure 9: Impulse Response Functions of the No-Arbitrage Deviation Measure andDealer Net Position to Shocks in the VIX index and Dealer Net Position
0 5 10 15 20 25
Weeks
-0.30
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co
re
IRF of DEV to VIX Shock
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IRF of DEV to DNP Shock
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IRF of DNP to VIX Shock
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IRF of DNP to DEV Shock
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IRF of VIX to DNP Shock
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Weeks
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0.00
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Z-S
co
re
IRF of VIX to DEV Shock
This figure reports the impulse response functions (IRFs) from the VAR. The IRFs are identified froma Cholesky decomposition with the ordering: VIX, DNP, DEV. The VAR is estimated using weeklydata from 2010 to 2018 (T = 469). The IRFs in the first row show that DEV is decreasing (increasing)in VIX (DNP) shocks. The IRFs in the second row show that the DNP is decreasing (increasing) inVIX (DEV) shocks. The IRFs in the third row show that the VIX exhibits little response to DEVor DNP shocks, except for a short term-response to DEV shocks that becomes insignificant after afew weeks. The 95% pointwise confidence intervals in gray are block bootstrapped.
53
Figure 10: Event Study of Deviation Measure Around VIX Futures Margin Changes,FOMC days, Nonfarm Payrolls, Month-End, and Quarter-End Dates
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.50
-0.25
0.00
0.25
0.50C
ha
ng
e in
De
via
tio
n
-2.00
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4.00
6.00
Cha
ng
e in
VIX
Ind
ex
Margin Increase
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
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an
ge
in
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n
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ng
e in
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ex
Margin Decrease
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
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an
ge
in
De
via
tio
n
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Change in V
IX In
de
x
FOMC
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
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-0.25
0.00
0.25
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an
ge
in
De
via
tio
n
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Change in V
IX In
de
x
Non-Farm Payrolls
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.50
-0.25
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Ch
an
ge
in
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n
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Change in V
IX I
nd
ex
Month-End
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
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0.00
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an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
Change in V
IX I
nd
ex
Quarter-End
This figure plots the event time reaction of the average deviation measure and VIX index to changesin the initial margin for the front month VIX futures contract, FOMC announcements, non-farmpayrolls announcements, end-of-month dates, and end-of-quarter dates. The figure reports thechange in the VIX index in red and the change in the average deviation measure across the frontsix contracts in blue with a 95% pointwise confidence interval in gray for the ten days before andafter the event.
54
Appendix:
“The Law of One Price in Equity Volatility Markets”
Peter Van Tassel
December 2019
A Appendix
Table A.1: Correlation of Alternative Deviation Measures
This table presents the correlation of the baseline deviation measure with alternative specifications.Panel A considers different ways of removing the bias from the upper bound, using a regression-based model (A.I) or term-structure model (A.II). Panel B considers alternative data sources. B.Iuses VIX settlement prices that are not synchronized with SPX option quotes during the later yearsin the sample. B.II through B.V use alternative data sources for computing variance swap forwardrates. The columns report the correlation for different contracts and for the average deviation acrosscontracts. Overall, the alternative measures are highly correlated with the baseline measure.
Correlation of Baseline Deviation Measure with Alternative MeasuresContract (n) (1) (2) (3) (4) (5) (6) Avg.
Panel A: Convexity Adjustments
Panel A.I: Regression-based convexity adjustment (Mar04-Dec18)Correlation in levels 1.00 1.00 1.00 1.00 1.00 1.00 1.00Correlation in monthly changes 1.00 1.00 1.00 1.00 1.00 0.99 1.00Correlation in weekly changes 1.00 1.00 1.00 1.00 1.00 1.00 1.00
Panel A.II: Term-structure model convexity adjustment (Mar04-Dec18)Correlation in levels 0.99 0.97 0.97 0.97 0.97 0.97 0.96Correlation in monthly changes 0.99 0.99 0.99 0.99 0.99 0.99 0.99Correlation in weekly changes 0.99 0.99 0.99 0.99 0.99 1.00 0.98
Panel B: Alternative Data Sources
Panel B.I: VIX settlement prices, not synchronized (Mar04-Dec18)Correlation in levels 0.96 0.96 0.98 0.99 0.99 0.99 0.98Correlation in monthly changes 0.95 0.94 0.96 0.97 0.97 0.98 0.95Correlation in weekly changes 0.94 0.90 0.93 0.94 0.95 0.97 0.89
Panel B.II: Bloomberg data for VS forward (Nov08-Dec18)Correlation in levels 0.66 0.69 0.71 0.77 0.85 0.89 0.87Correlation in monthly changes 0.72 0.63 0.64 0.72 0.70 0.78 0.81Correlation in weekly changes 0.69 0.48 0.66 0.66 0.67 0.73 0.72
Panel B.III: CBOE VIX, VIX3M, VIX6M for VS forward (Nov08-Dec18)Correlation in levels 0.68 0.61 0.85 0.85 0.88 0.86 0.93Correlation in monthly changes 0.74 0.43 0.80 0.80 0.81 0.74 0.89Correlation in weekly changes 0.71 0.15 0.80 0.69 0.70 0.58 0.80
Panel B.IV: Hedge Fund OTC quotes for VS forward (Mar04-Nov13)Correlation in levels 0.41 0.53 0.68 0.82 0.83 0.77 0.71Correlation in monthly changes 0.46 0.41 0.51 0.66 0.65 0.66 0.53Correlation in weekly changes 0.26 0.27 0.40 0.51 0.51 0.62 0.27
Panel B.V: Markit OTC quotes for VS forward (monthly Sep06-Dec15)Correlation in Levels 0.64 0.74 0.80 0.85 0.75 0.79 0.87Correlation in monthly changes 0.67 0.72 0.74 0.82 0.81 0.82 0.88
1
Table A.2: Hedge Ratios for Return Predictability Regressions
This table reports the hedge ratios and explanatory power from the first stage in the return pre-dictability regressions for Tables 5, 6, and 7. The hedge ratios are significant across contracts andsample periods. Compared to the stock market, variance swap forwards provide more explanatorypower for VIX futures returns, particularly for the longer-dated contracts.
Hedge Ratios for Weekly Return Predictability Regressions
Contract (n) (1) (2) (3) (4) (5) (6)
Panel A: VIX Futures onto Variance Swap Forwards (Table 5)
Panel A.I: Sample from 2004 to 2018βn 10.36∗∗∗ 14.23∗∗∗ 14.48∗∗∗ 14.31∗∗∗ 14.98∗∗∗ 14.67∗∗∗
(1.50) (1.16) (1.14) (1.14) (0.78) (0.87)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.574 0.770 0.757 0.757 0.715 0.664
Panel A.II: Sample from 2010 to 2018βn 15.28∗∗∗ 21.04∗∗∗ 19.65∗∗∗ 18.53∗∗∗ 18.09∗∗∗ 16.64∗∗∗
(1.69) (0.83) (0.59) (0.54) (0.62) (0.51)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.627 0.845 0.830 0.824 0.793 0.760
Panel B: VIX Futures onto Stock Market (Table 6)
Panel B.I: Sample from 2004 to 2018βn 0.73∗∗∗ 0.58∗∗∗ 0.45∗∗∗ 0.38∗∗∗ 0.34∗∗∗ 0.30∗∗∗
(0.04) (0.03) (0.03) (0.02) (0.02) (0.02)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.574 0.566 0.551 0.535 0.519 0.496
Panel B.II: Sample from 2010 to 2018βn 0.84∗∗∗ 0.70∗∗∗ 0.56∗∗∗ 0.47∗∗∗ 0.41∗∗∗ 0.37∗∗∗
(0.05) (0.03) (0.02) (0.02) (0.02) (0.02)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.592 0.676 0.669 0.651 0.620 0.606
Panel C: Variance Swap Forwards onto Stock Market (Table 7)
Panel C.I: Sample from 2004 to 2018βn 0.05∗∗∗ 0.03∗∗∗ 0.02∗∗∗ 0.02∗∗∗ 0.02∗∗∗ 0.02∗∗∗
(0.01) (0.00) (0.00) (0.00) (0.00) (0.00)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.546 0.451 0.388 0.411 0.444 0.403
Panel C.II: Sample from 2010 to 2018βn 0.05∗∗∗ 0.03∗∗∗ 0.02∗∗∗ 0.02∗∗∗ 0.02∗∗∗ 0.02∗∗∗
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.642 0.650 0.609 0.600 0.600 0.568
Newey-West SEs with 3 · h lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
2
Table A.3: Predicting VIX Futures Hedged Returns with LOOP Deviations:Robustness to Sample Period and Forecast Horizon
This table reports the predictability of the deviation measure for VIX futures returns hedged withvariance swap forward returns over one-day, one-week, and one-month horizons across sample peri-ods.
Return Predictability Regressions: RV IXt+h,n − βnRV St+h,n = γ′nxt,n + εt+h,nContract (n) (1) (2) (3) (4) (5) (6)
Panel A.I: Sample period 2004-2018, daily returns h = 1Deviation 0.21∗∗∗ 0.17∗∗∗ 0.21∗∗∗ 0.18∗∗∗ 0.27∗∗∗ 0.27∗∗∗
(0.04) (0.03) (0.04) (0.02) (0.04) (0.05)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.044 0.030 0.042 0.031 0.073 0.072
Panel A.II: Sample period 2004-2018, weekly returns h = 5Deviation 0.19∗∗∗ 0.23∗∗∗ 0.27∗∗∗ 0.25∗∗∗ 0.36∗∗∗ 0.38∗∗∗
(0.03) (0.05) (0.05) (0.03) (0.07) (0.08)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.038 0.051 0.075 0.063 0.129 0.145
Panel A.III: Sample period 2004-2018, monthly returns h = 21Deviation 0.26∗∗∗ 0.24∗∗∗ 0.29∗∗∗ 0.29∗∗∗ 0.41∗∗∗ 0.45∗∗∗
(0.09) (0.06) (0.05) (0.05) (0.08) (0.07)Observations 3344 3685 3685 3631 3194 3187Adjusted R2 0.068 0.059 0.084 0.084 0.167 0.198
Panel B.I: Sample period 2010-2018, daily returns h = 1Deviation 0.22∗∗∗ 0.19∗∗∗ 0.17∗∗∗ 0.17∗∗∗ 0.30∗∗∗ 0.26∗∗∗
(0.06) (0.04) (0.03) (0.03) (0.03) (0.03)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.046 0.037 0.030 0.027 0.092 0.068
Panel B.II: Sample period 2010-2018, weekly returns h = 5Deviation 0.23∗∗∗ 0.34∗∗∗ 0.28∗∗∗ 0.26∗∗∗ 0.40∗∗∗ 0.38∗∗∗
(0.04) (0.03) (0.04) (0.04) (0.05) (0.05)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.052 0.112 0.078 0.068 0.158 0.143
Panel B.III: Sample period 2010-2018, monthly returns h = 21Deviation 0.14∗∗∗ 0.42∗∗∗ 0.44∗∗∗ 0.36∗∗∗ 0.51∗∗∗ 0.51∗∗∗
(0.05) (0.04) (0.07) (0.07) (0.06) (0.06)Observations 2019 2233 2233 2233 2233 2233Adjusted R2 0.019 0.174 0.196 0.133 0.255 0.256Newey-West SEs with 3 · h lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
3
Table A.4: Predicting VIX Futures Hedged Returns with LOOP Deviations:Robustness to Deviation Measure
This table reports the predictability of the deviation measure for VIX futures returns hedged withvariance swap forward returns over a weekly horizon (h=5) using different deviation measures. Thesample period varies depending on when the different data sources are available.
Return Predictability Regressions: RV IXt+h,n − βnRV St+h,n = γ′nxt,n + εt+h,nContract (n) (1) (2) (3) (4) (5) (6)Panel A: Deviation from paper (h = 5, Mar04-Dec18, daily overlapping)Deviation 0.19∗∗∗ 0.23∗∗∗ 0.27∗∗∗ 0.25∗∗∗ 0.36∗∗∗ 0.38∗∗∗
(0.03) (0.05) (0.05) (0.03) (0.07) (0.08)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.038 0.051 0.075 0.063 0.129 0.145Panel B: Deviation 1-day lag (h = 5, Mar04-Dec18, daily overlapping)Deviation 1d Lag 0.16∗∗∗ 0.13∗∗∗ 0.16∗∗∗ 0.12∗∗∗ 0.20∗∗∗ 0.20∗∗∗
(0.06) (0.04) (0.03) (0.02) (0.04) (0.05)Observations 3355 3682 3695 3636 3205 3037Adjusted R2 0.011 0.022 0.049 0.041 0.105 0.108Panel C: Deviation 5-day moving average (h = 5, Mar04-Dec18, daily overlapping)Deviation 5d MA 0.17∗∗ 0.14∗∗∗ 0.16∗∗∗ 0.11∗∗∗ 0.20∗∗∗ 0.19∗∗∗
(0.07) (0.05) (0.03) (0.02) (0.04) (0.05)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.012 0.025 0.050 0.035 0.106 0.097Panel D: Regression-based convexity adjustment (h = 5, Mar04-Dec18, daily overlapping)Deviation - Regression Adj. 0.29∗∗∗ 0.20∗∗∗ 0.19∗∗∗ 0.15∗∗∗ 0.22∗∗∗ 0.23∗∗∗
(0.05) (0.04) (0.03) (0.02) (0.04) (0.05)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.037 0.051 0.073 0.061 0.122 0.135Panel E: Term-structure model convexity adjustment (h = 5, Mar04-Dec18, daily overlapping)Deviation - Term-Structure Adj. 0.29∗∗∗ 0.19∗∗∗ 0.17∗∗∗ 0.13∗∗∗ 0.23∗∗∗ 0.23∗∗∗
(0.05) (0.04) (0.03) (0.02) (0.05) (0.05)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.036 0.046 0.061 0.047 0.134 0.134Panel F: Bloomberg data (h = 5, Nov08-Dec18, daily overlapping)Deviation Bloomberg Data 0.34∗∗∗ 0.18∗∗∗ 0.12∗∗∗ 0.08∗∗∗ 0.26∗∗∗ 0.20∗∗∗
(0.08) (0.05) (0.03) (0.03) (0.05) (0.05)Observations 2267 2511 2511 2511 2511 2511Adjusted R2 0.047 0.037 0.027 0.017 0.170 0.107Panel G: VIX, VIX3M, VIX6M Indices (h = 5, Nov08-Dec18, daily overlapping)Deviation VIX Indices 0.25∗∗∗ 0.14∗∗∗ 0.13∗∗∗ 0.12∗∗∗ 0.24∗∗∗ 0.17∗∗∗
(0.07) (0.05) (0.04) (0.03) (0.03) (0.04)Observations 2266 2510 2510 2510 2510 2510Adjusted R2 0.024 0.023 0.034 0.033 0.147 0.083Panel H: Hedge Fund data (h = 5, Mar04-Nov13, daily overlapping)Deviation Hedge Fund Data 0.12 0.08 0.02 0.07∗∗∗ 0.14∗∗∗ 0.15∗∗∗
(0.08) (0.05) (0.06) (0.03) (0.04) (0.05)Observations 2209 2429 2429 2375 1938 1931Adjusted R2 0.007 0.008 0.001 0.013 0.041 0.045Panel I: Markit data (h = 21, Sep06-Dec15, end-of-month data, non-overlapping)Deviation Markit Data 0.51∗∗ 0.11 0.06 0.11 0.28∗∗ 0.32∗∗
(0.23) (0.27) (0.11) (0.09) (0.14) (0.14)Observations 108 108 108 108 108 108Adjusted R2 0.019 -0.004 -0.006 0.009 0.076 0.117Newey-West SEs with 3 · h lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
4
Table A.5: Predicting VIX Futures Hedged Returns with LOOP Deviations:Robustness to Return Definition - Percentage Returns
This table reports the predictability of the deviation measure for VIX futures returns hedged withvariance swap forward returns over a weekly horizon (h=5) where returns are defined as the per-centage return rather than the change in the futures price or change in the forward price. Thedeviation measure remains significant in these regressions. As noted in the paper, capturing per-centage returns for VIX futures and variance swap forwards requires a dynamic trading strategythat may entail additional transaction costs in practice.
Predictability Regression: RV IXt+h,n − βnRV St+h,n = γ′nxt,n + εt+h,n
Return Definitions: RV IXt+h,n ≡Futt,n−Futt+h,n
Futt,n, RV SFt+h,n ≡
Ft,n−Ft+h,n
Ft,n
Contract (n) (1) (2) (3) (4) (5) (6)
Panel A.I: Full sample from 2004 to 2018Deviation 0.19∗∗∗ 0.34∗∗∗ 0.37∗∗∗ 0.33∗∗∗ 0.32∗∗∗ 0.35∗∗∗
(0.03) (0.03) (0.03) (0.03) (0.05) (0.05)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.036 0.114 0.140 0.112 0.101 0.122
Panel A.II: Full sample from 2004 to 2018 with controlsDeviation 0.23∗∗∗ 0.42∗∗∗ 0.43∗∗∗ 0.38∗∗∗ 0.30∗∗∗ 0.37∗∗∗
(0.04) (0.04) (0.03) (0.04) (0.05) (0.05)VIX 0.22∗∗ 0.14∗ 0.17∗ 0.10 0.01 0.20∗∗
(0.09) (0.08) (0.09) (0.08) (0.10) (0.10)RV -0.17∗∗ -0.05 0.00 0.06 -0.13 -0.19∗
(0.08) (0.07) (0.09) (0.08) (0.11) (0.10)RMRF -0.02 -0.09∗∗∗ -0.03 -0.09∗∗∗ -0.02 0.01
(0.02) (0.03) (0.04) (0.03) (0.04) (0.04)VLM -0.04 0.14∗∗∗ 0.16∗∗∗ 0.06∗ 0.13∗∗∗ 0.17∗∗∗
(0.03) (0.04) (0.03) (0.03) (0.03) (0.04)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.045 0.150 0.187 0.150 0.139 0.154
Panel B.I: Post-crisis sample from 2010 to 2018Deviation 0.14∗∗∗ 0.29∗∗∗ 0.30∗∗∗ 0.31∗∗∗ 0.37∗∗∗ 0.37∗∗∗
(0.04) (0.03) (0.04) (0.04) (0.05) (0.05)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.019 0.081 0.090 0.096 0.135 0.133
Panel B.II: Post-crisis sample from 2010 to 2018 with controlsDeviation 0.21∗∗∗ 0.37∗∗∗ 0.35∗∗∗ 0.33∗∗∗ 0.33∗∗∗ 0.37∗∗∗
(0.03) (0.05) (0.04) (0.05) (0.05) (0.05)VIX 0.21∗∗∗ 0.21∗∗∗ 0.20∗∗ 0.04 -0.10 0.02
(0.08) (0.07) (0.09) (0.09) (0.10) (0.10)RV -0.08 -0.07 0.04 0.10 -0.06 -0.02
(0.07) (0.06) (0.08) (0.08) (0.09) (0.09)RMRF -0.03 -0.07∗∗ -0.03 -0.07 -0.03 0.01
(0.03) (0.04) (0.04) (0.04) (0.05) (0.05)VLM -0.03 0.11∗∗ 0.17∗∗∗ -0.00 -0.01 0.09∗∗
(0.04) (0.05) (0.03) (0.05) (0.04) (0.04)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.038 0.127 0.173 0.120 0.154 0.140Newey-West SEs with 15 lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
5
Table A.6: Predicting VIX Futures Hedged Returns with LOOP Deviations:Robustness to Return Definition - Log Returns
This table reports the predictability of the deviation measure for VIX futures returns hedged withvariance swap forward returns over a weekly horizon (h=5) where returns are defined as log-returnsrather than the change in the futures price or change in the forward price. The deviation measureremains significant in these regressions. As noted in the paper, capturing log-returns for VIXfutures and variance swap forwards requires a dynamic trading strategy that may entail additionaltransaction costs in practice.
Predictability Regression: RV IXt+h,n − βnRV St+h,n = γ′nxt,n + εt+h,n
Return Definition: RV IXt+h,n = ln(Futt,n/Futt+h,n), RV SFt+h,n ≡ ln(Ft,n/Ft+h,n)
Contract (n) (1) (2) (3) (4) (5) (6)
Panel A.I: Full sample from 2004 to 2018Deviation 0.27∗∗∗ 0.42∗∗∗ 0.41∗∗∗ 0.35∗∗∗ 0.33∗∗∗ 0.35∗∗∗
(0.03) (0.04) (0.03) (0.03) (0.05) (0.05)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.074 0.174 0.169 0.125 0.108 0.124
Panel A.II: Full sample from 2004 to 2018 with controlsDeviation 0.28∗∗∗ 0.49∗∗∗ 0.46∗∗∗ 0.39∗∗∗ 0.30∗∗∗ 0.38∗∗∗
(0.03) (0.04) (0.03) (0.04) (0.05) (0.05)VIX 0.16∗ 0.13 0.16 0.11 -0.02 0.19∗
(0.09) (0.09) (0.10) (0.09) (0.10) (0.11)RV -0.11 -0.02 -0.01 0.03 -0.13 -0.17∗
(0.08) (0.09) (0.10) (0.08) (0.10) (0.10)RMRF 0.00 -0.07∗∗ -0.03 -0.07∗∗ -0.01 0.02
(0.03) (0.03) (0.04) (0.03) (0.04) (0.04)VLM -0.11∗∗∗ 0.11∗∗ 0.16∗∗∗ 0.05 0.13∗∗∗ 0.17∗∗∗
(0.03) (0.04) (0.03) (0.03) (0.03) (0.04)Observations 3356 3698 3698 3644 3207 3200Adjusted R2 0.090 0.201 0.213 0.156 0.149 0.155
Panel B.I: Post-crisis sample from 2010 to 2018Deviation 0.23∗∗∗ 0.40∗∗∗ 0.35∗∗∗ 0.33∗∗∗ 0.38∗∗∗ 0.37∗∗∗
(0.04) (0.03) (0.04) (0.04) (0.05) (0.05)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.055 0.160 0.119 0.112 0.145 0.137
Panel B.II: Post-crisis sample from 2010 to 2018 with controlsDeviation 0.26∗∗∗ 0.48∗∗∗ 0.40∗∗∗ 0.36∗∗∗ 0.34∗∗∗ 0.37∗∗∗
(0.03) (0.05) (0.04) (0.05) (0.05) (0.06)VIX 0.15∗ 0.22∗∗ 0.19∗ 0.02 -0.13 0.01
(0.08) (0.11) (0.10) (0.09) (0.10) (0.10)RV -0.05 -0.06 0.07 0.12 -0.05 -0.01
(0.08) (0.08) (0.09) (0.09) (0.10) (0.09)RMRF -0.00 -0.05 -0.02 -0.07 -0.02 0.01
(0.04) (0.04) (0.04) (0.04) (0.04) (0.05)VLM -0.08∗∗ 0.07 0.19∗∗∗ -0.00 -0.01 0.09∗∗
(0.04) (0.07) (0.04) (0.05) (0.04) (0.04)Observations 2030 2246 2246 2246 2246 2246Adjusted R2 0.072 0.197 0.210 0.133 0.169 0.144Newey-West SEs with 15 lags in parentheses, ∗ p<.10, ∗∗ p<.05, ∗∗∗ p<.01
6
Figure A.1: Examples of VIX Futures Law of One Price Deviations
0 1 2 3 4 5 6 7 8 9 10
Contract Number
20
22
24
26
28
30
Price
Futures Prices vs. No-Arbitrage Bounds on 02/27/12
Futt,n
Settlement
Futt,n
4pm Price
Fwdt,n
Synthetic Upper Bound
Fwdt,n
Bloomberg Upper Bound
Fwdt,n
VIX Index Upper Bound
Fwdt,n
Hedge Fund Upper Bound
0 1 2 3 4 5 6 7 8 9 10
Contract Number
26
28
30
32
34
36
38
40
42
44
Price
Futures Prices vs. No-Arbitrage Bounds on 08/10/11
Futt,n
Settlement
Futt,n
4pm Price
Fwdt,n
Synthetic Upper Bound
Fwdt,n
Bloomberg Upper Bound
Fwdt,n
VIX Index Upper Bound
Fwdt,n
Hedge Fund Upper Bound
Fvolt,n
Synthetic Lower Bound
This figure compares VIX futures prices at 4pm and at 4:15pm (settlement) to variance swap forwardrates estimated from a variety of alternative data sources on two different days. The results illustratethe robustness of Figure 3. Regardless of which of the four data sources are used to compute varianceswap forward rates or which of the two VIX futures prices are used, the top plot features examplesof static arbitrage opportunities in which the prices of VIX futures are above their non-parametric,no-arbitrage upper bounds. The bottom plot illustrates the opposite case where the prices of VIXfutures are well below their upper bounds and are even below an estimate of the lower bounds.
7
Figure A.2: Deviation Measure Computed from Alternative Datasets
04 05 06 07 08 09 10 11 12 13 14 15 16 17 18
Date
-5
-4
-3
-2
-1
0
1
2
Avera
ge D
evia
tion (
5d M
A)
Deviation from Synthetic VS Rates (Paper)
Deviation from Bloomberg Data = 0.89
Deviation from CBOE VIX Indices = 0.95
Deviation from Hedge Fund OTC = 0.82
This figure plots the main deviation measure from the paper against alternative deviation measuresthat are computed by estimating variance swap forward rates from different datasets. For eachdeviation measure, the figure reports the average deviation across the front six contracts as a five-day moving average. The deviation measure from the paper is highly correlated with the alternativedeviation measures. The correlation is reported in the legend over the sample period when both themain deviation measure and alternative measure are available. As before, positive values are law ofone price violations. Gray shading indicates NBER recessions.
8
Figure A.3: Deviation Measure By VIX Futures Contract
04 06 08 10 12 14 16 18 0
10
20
30
40
50
60
70
Price
(5
d M
A)
Contract 1
Futures Price
Forward Price
04 06 08 10 12 14 16 18-8
-6
-4
-2
0
2
4
De
via
tion
(5
d M
A)
Contract 1
04 06 08 10 12 14 16 18 0
10
20
30
40
50
60
Price
(5
d M
A)
Contract 2
04 06 08 10 12 14 16 18-8
-6
-4
-2
0
2
4
De
via
tion
(5
d M
A)
Contract 2
04 06 08 10 12 14 16 18 0
10
20
30
40
50
60
Price
(5
d M
A)
Contract 3
04 06 08 10 12 14 16 18-8
-6
-4
-2
0
2
4
De
via
tion
(5
d M
A)
Contract 3
04 06 08 10 12 14 16 18 0
10
20
30
40
50
60
Price (
5d M
A)
Contract 4
04 06 08 10 12 14 16 18-8
-6
-4
-2
0
2
4
Devia
tio
n (
5d M
A)
Contract 4
04 06 08 10 12 14 16 18 0
10
20
30
40
50
60
Price (
5d M
A)
Contract 5
04 06 08 10 12 14 16 18-8
-6
-4
-2
0
2
4
Devia
tio
n (
5d M
A)
Contract 5
04 06 08 10 12 14 16 18 0
10
20
30
40
50
60
Price (
5d M
A)
Contract 6
04 06 08 10 12 14 16 18-8
-6
-4
-2
0
2
4
Devia
tion (
5d M
A)
Contract 6
This figure plots the VIX futures price and variance swap forward rate (upper bound) on the leftand deviation measure, or difference between the futures price and forward price, on the right foreach contract. The prices and deviation measure for the longer-dated contracts become availablelater in the sample, motivating the 2007-2018 and 2010-2018 sample periods that are used in someof the regression and summary statistics analysis to provide a balanced panel across contracts.
9
Figure A.4: Distribution of VIX Futures Prices Relative tothe No-Arbitrage Upper Bound from 2007 to 2018
This figure plots histograms and kernel density estimates of the law of one price deviation measureby VIX futures contract from 2007 to 2018. The distribution is negatively skewed for the frontcontract and second contract. The histograms indicate the presence of law of one price violationsfrom the probability mass for the deviation measure being greater than zero which correspondsto VIX futures prices being greater than the upper bound. The histograms also reveal how thedeviation measures exhibit large negative values that may also represent law of one price violationsto the extent that VIX futures prices go below volatility swap forward rates. The lower boundviolations cannot be measured directly from these histograms, but are reported in the next figure.
10
Figure A.5: Distribution of VIX Futures Prices Relative tothe No-Arbitrage Lower Bound from 2007 to 2018
This figure plots histograms and kernel density estimates of VIX futures prices relative to their law ofone price lower bound from 2007 to 2018. The lower bound is computed as Fwdt,n−(UBt,n−LBt,n)
where UBt,n is the variance swap forward rate and LBt,n is the volatility swap forward rate estimatedfrom a no-arbitrage term-structure model on day t for the n-th contract following the approach inVan Tassel (2019). In this case, the histograms indicate the presence of law of one price violationsfrom the probability mass below zero which corresponds to cases in which the VIX futures price isbelow the lower bound. Lower bound violations are more pronounced for the longer-dated contractsconsistent with Table 3.
11
Figure A.6: Impulse Response Functions of Deviation and Dealer Position toVIX and Dealer Position Shocks Across Trivariate VAR Specifications
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
reIRF of Deviation to VIX Shock:
Trivariate VAR with Cholesky Decomposition
Ordered: VIX, DNP, DEV
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of Deviation to Dealer Net Position Shock:
Trivariate VAR with Cholesky Decomposition
Ordered: VIX, DNP, DEV
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
Trivariate VAR with Cholesky Decomposition
Ordered: VIX, DEV, DNP
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
Trivariate VAR with Cholesky Decomposition
Ordered: VIX, DEV, DNP
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
Trivariate VAR with Cholesky Decomposition
Ordered: DNP, VIX, DEV
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
Trivariate VAR with Cholesky Decomposition
Ordered: DNP, VIX, DEV
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
Trivariate VAR with Spectral Decomposition
0 5 10 15 20 25
Weeks
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
Trivariate VAR with Spectral Decomposition
This figure plots the impulse response function of of the no-arbitrage deviation and dealer positionto VIX and dealer position shocks across different trivariate VAR specifications. The first three rowsreport different orderings of the variables for a Cholesky decomposition. The fourth row reports theIRFs for a spectral decomposition. The variables in the vector autoregression are yt = [DeviationtV IXt Dealer Positiont]. The VAR is estimated using weekly data from 2010 to 2018 (T = 469).Across specifications, the IRFs take on similar shapes. The deviation measure declines in responseto a risk shock and increases in response to a demand shock. The impact of the risk shock dies outafter a few weeks whereas the demand shocks are more persistent.
12
Figure A.7: Deviation Measure versus Risk and Demand Variables
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-S
co
re
Deviation vs. Stock Market Returns: Correlation = 33%
Average Deviation
RMRF (1mn MA)
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-S
co
re
Deviation vs. CoT Dealer Net Position: Correlation = 57%
Average Deviation
CoT Dealer Net Position
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-S
co
re
Deviation vs. Realized Variance: Correlation = -28%
Average Deviation
RV
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-S
co
re
Deviation vs. VIX Short-Term ETP Demand: Correlation = 51%
Average Deviation
VIX Short-Term ETP Demand
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-S
co
re
Deviation vs. VIX Index: Correlation = -24%
Average Deviation
VIX Index
10 11 12 13 14 15 16 17 18 19
Date
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-S
co
re
Deviation vs. VIX Options Customer Delta: Correlation = 37%
Average Deviation
VIX Options Customer Delta (1mn MA)
This figure plots the deviation measure against different risk and demand variables. The riskvariables include stock market returns, realized variance, and the VIX Index. Increases in risk asmeasured by negative stock market returns or increases in volatility are negatively correlated withthe deviation measure. The demand variables are Dealer Position from the CoT Report, VIX ETPdemand, and VIX options customer delta. The demand variables are positively correlated with thedeviation measure. The sample period is 2010 to 2018 using weekly data (T = 469).
13
Figure A.8: Impulse Response Functions of Deviation Measureto Risk and Demand Shocks in Bivariate VARs
0 5 10 15 20 25
Weeks
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Z-S
co
re
IRF of VIX Futures Deviation to Risk Shocks:
RMRF (1mn MA)
0 5 10 15 20 25
Weeks
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Z-S
co
re
IRF of VIX Futures Deviation to Demand Shocks:
Dealer Net Position
0 5 10 15 20 25
Weeks
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Z-S
co
re
Realized Variance (RV)
0 5 10 15 20 25
Weeks
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Z-S
co
re
VIX ETP Demand
0 5 10 15 20 25
Weeks
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Z-S
co
re
VIX Index
0 5 10 15 20 25
Weeks
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Z-S
co
re
VIX Options Customer Delta (1mn MA)
This figure plots impulse response functions from bivariate vector autoregressions to illustrate howthe deviation measure reacts to risk and demand shocks. The left column reports the IRFs frombivariate VARs with risk variables. The right column reports the IRFs from bivariate VARs withdemand variables. The IRFs are from a Cholesky decomposition with the deviation measure orderedsecond. The 95% confidence intervals in gray are block bootstrapped. The lag length is selectedusing the SBIC criterion. Similar to the trivariate VAR discussed in the paper and the time-seriesplots, the deviation measure decreases when risk increases and increases when demand increases.The magnitude of the response is larger for the risk shocks, but more persistent for the dealerposition and VIX ETP demand shocks.
14
Figure A.9: Impulse Response Function of Deviation Measure Across Contracts
0 10 20
Weeks
-0.20
0.00
0.20
0.40
0.60
Z-S
co
re
IRF of D1 to D1
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D1 to D2
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D1 to D3
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D1 to D4
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D2 to D1
0 10 20
Weeks
-0.20
0.00
0.20
0.40
0.60
Z-S
co
re
IRF of D2 to D2
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D2 to D3
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D2 to D4
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D3 to D1
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D3 to D2
0 10 20
Weeks
-0.20
0.00
0.20
0.40
0.60
Z-S
co
re
IRF of D3 to D3
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D3 to D4
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D4 to D1
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D4 to D2
0 10 20
Weeks
-0.10
0.00
0.10
0.20
0.30
Z-S
co
re
IRF of D4 to D3
0 10 20
Weeks
-0.20
0.00
0.20
0.40
0.60
Z-S
co
re
IRF of D4 to D4
This figure reports the IRFs from a weekly VAR with yt = [Devt,1 Devt,2 Devt,3 Devt,4] from 2010to 2018. The goal of these IRFs is to highlight how shocks to different contracts propagate to othercontracts. The results indicate that shocks to the front contract correspond to increases in thesecond and third contract that die out after a few months. Shocks to the fourth contract resultin delayed increases in the second and third contract. The IRFs are computed using a Choleskydecomposition with the variables ordered as the deviation for the front, second, third, and fourthcontract respectively. The results are reported using weekly data for comparison to the other IRFsreported in the paper and Appendix and focus on the front four contracts for parsimony. Similarqualitative results hold for daily data and for the front six contracts.
15
Figure A.10: Event Study of VIX Futures Deviation AroundLarge Risk and Demand Shocks
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
Ch
an
ge
in V
IX I
nde
x
RMRF Positive (Top 50 days)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
Ch
an
ge
in V
IX I
nde
x
RMRF Negative (Top 50 days)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
Ch
an
ge
in V
IX I
nde
x
VIX Increase (Top 50 days)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
Ch
an
ge
in V
IX I
nde
x
VIX Decrease (Top 50 days)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.50
-0.25
0.00
0.25
0.50
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
Change in V
IX Index
Dealer Position Increase (Top 50 weeks)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.50
-0.25
0.00
0.25
0.50
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
Change in V
IX Index
Dealer Position Decrease (Top 50 weeks)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.50
-0.25
0.00
0.25
0.50
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
Change in V
IX Index
VIX Opt Cust Delta Increase (Top 50 days)
-10 -8 -6 -4 -2 0 2 4 6 8 10
Event Date
-0.50
-0.25
0.00
0.25
0.50
Ch
an
ge
in
De
via
tio
n
-2.00
0.00
2.00
4.00
6.00
Change in V
IX Index
VIX Opt Cust Delta Decrease (Top 50 days)
This figure plots the event time reaction of the deviation measure and VIX index to the largestincreases and decreases in stock market returns, the VIX index, dealer position, and VIX optionscustomer delta throughout the sample. The figure reports the change in the VIX index and averagedeviation measure across the front six contracts in the ten days before and after the event.
16
Figure A.11: VIX Futures Trading Strategy: Position and Post-Crisis Sharpe Ratios
04 05 06 07 08 09 10 11 12 13 14 15 16 17 18
Date
-4
-3
-2
-1
0
1
2
Devia
tion (
5d M
A)
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Positio
n (
5d M
A)
VIX Futures Law of One Price Deviations vs. Trading Strategy Position: Correlation = -0.61
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0
0.2
0.4
0.6
0.8
1
Sh
arp
e R
atio
(A
nn
ua
lize
d)
Contract 1: 2010-2018
VS Fwd Hedge
Stock Market Hedge
Stock Market Hedge with T-Costs
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Sh
arp
e R
atio
(A
nn
ua
lize
d)
Contracts 1-2: 2010-2018
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0
0.5
1
1.5
2
2.5
3
Sh
arp
e R
atio
(A
nn
ua
lize
d)
Contracts 1-4: 2010-2018
0 0.5 1 1.5 2 2.5 3
Trading Threshold (Deviation z-score)
0
0.5
1
1.5
2
2.5
3
3.5
Sh
arp
e R
atio
(A
nn
ua
lize
d)
Contracts 1-6: 2010-2018
The top figure plots the number of net positions in the relative value trading strategy against thedeviation measure over time. A long position corresponds to buying VIX futures that are hedgedwith variance swap forwards. Since the strategy trades the front six contracts, the number of netpositions is bounded between -6 and 6. The plot illustrates the negative correlation between thedeviation measure and the number of net positions. When the deviation measure is low (high),the strategy tends to buy (sell) VIX futures that are hedged with variance swap forwards. Thebottom plot reports the SRs for the different strategies varying the number of contracts traded andthreshold. This is analogous to Figure 7 but for the 2010 to 2018 post-crisis period.
17
